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# [https://oeis.org/A030740 A030740] (inverse) has a strange <code>Offset 2,1</code> which should be 1,1 IMHO as for the similiar [https://oeis.org/A030730 A030730], [https://oeis.org/A030750 A030750]. In the description there is a typo <code>j'2</code>. | # [https://oeis.org/A030740 A030740] (inverse) has a strange <code>Offset 2,1</code> which should be 1,1 IMHO as for the similiar [https://oeis.org/A030730 A030730], [https://oeis.org/A030750 A030750]. In the description there is a typo <code>j'2</code>. | ||
# A030709 = A030759, A030729 = A030779, A030749 = A030789, A030711 = A030761 | # A030709 = A030759, A030729 = A030779, A030749 = A030789, A030711 = A030761 | ||
# [https://oeis.org/A055186 A055186] should not carry keyword <code>base</ | # [https://oeis.org/A055186 A055186] should not carry keyword <code>base</code>. | ||
# The example in [https://oeis.org/A055168 A055168] has a typo. | |||
==List of ''cumulative counting'' sequences== | ==List of ''cumulative counting'' sequences== | ||
The list of 116 sequences below describes the variations. The list has the following entries (which are explained in more detail in the Perl program, below): | The list of 116 sequences below describes the variations. The list has the following entries (which are explained in more detail in the Perl program, below): | ||
Revision as of 09:32, 30 April 2018
Clark Kimberling defined - for example - in A217760:
Write 0 followed by segments defined inductively as follows: each segment tells how many times each previously written integer occurs, in the order of first occurrence. This is Method A (adjective-before-noun pairs); for Method B (noun-before-adjective), see A055168. Example: Start with 0, followed by the adjective-noun pair 1,0; followed by adjective-noun pairs 2,0 then 1,1; etc. Writing the pairs vertically, the initial segments are 0.|.1.|.2 1.|.3 3 1.|.4 5 2 2.|.5 6 5 3 1 1.|.6 9 6 5 2 4 1.|.7 11 8 6 4 6 4 1 ..|.0.|.0 1.|.0 1 2.|.0 1 2 3.|.0 1 2 3 4 5.|.0 1 2 3 4 5 6.|.0 1 2 3 4 5 6 9
Depending on the starting value s(1), with variations of the rules and derivations, there are more than 120 related sequences in the OEIS.
Results and Problems
A Perl program and a Unix makefile were written in order to evaluate these sequences. Only a few problems were found:
- The treatment of zero attributes and noun zero is not always clearly described, and the methods differ considerably with this respect.
- Most b-files are rather short, though some sequences have quite a number of terms in common (e.g. 61, 82) before they start to differ.
- A030717 (row 1) does not correspond with the derived sequences A030718 (row 2) and A030719-A030726. A030717 initially had no zero terms, but in 2014 the description was changed, and the b-file now has zero terms. This should be changed back.
- A030740 (inverse) has a strange
Offset 2,1which should be 1,1 IMHO as for the similiar A030730, A030750. In the description there is a typoj'2. - A030709 = A030759, A030729 = A030779, A030749 = A030789, A030711 = A030761
- A055186 should not carry keyword
base. - The example in A055168 has a typo.
List of cumulative counting sequences
The list of 116 sequences below describes the variations. The list has the following entries (which are explained in more detail in the Perl program, below):
- the OEIS A-number,
- a major rule name (A, B were defined by C. Kimberling)
- the starting value s(1)
- a code for the ordering in the segements (increasing/decreasing order, first appearance)
- a code for the row number (1, 2, ...)
- optional parameters
A030707 A 1 io 1 A030708 A 1 io 2 A030709 T 1 io 1 A030710 I 1 io 1 A030711 J 1 io 1 A030712 K 1 io 1 A030713 P 1 io 1 parm=1 A030714 P 1 io 1 parm=2 A030715 P 1 io 1 parm=3 A030716 P 1 io 1 parm=4 A030717 A 1 io 5 with0=3 A030718 A 1 io 6 A030719 T 1 io 5 A030720 I 1 io 5 A030721 J 1 io 5 A030722 K 1 io 5 A030723 P 1 io 5 parm=1 A030724 P 1 io 5 parm=2 A030725 P 1 io 5 parm=3 A030726 P 1 io 5 parm=4 A030727 A 3 io 1 A030728 A 3 io 2 A030729 T 3 io 3 A030730 I 3 io 1 A030731 J 3 io 1 A030732 K 3 io 1 A030733 P 3 io 1 parm=1 A030734 P 3 io 1 parm=2 A030735 P 3 io 1 parm=3 A030736 P 3 io 1 parm=4 A030737 A 2 io 1 A030738 A 2 io 2 A030739 T 2 io 3 A030740 I 2 io 1 A030741 J 2 io 1 A030742 K 2 io 1 A030743 P 2 io 1 parm=1 A030744 P 2 io 1 parm=2 A030745 P 2 io 1 parm=3 A030746 P 2 io 1 parm=4 A030747 A 4 io 1 A030748 A 4 io 2 A030749 T 4 io 3 A030750 I 4 io 1 A030751 J 4 io 1 A030752 K 4 io 1 A030753 P 4 io 1 parm=1 A030754 P 4 io 1 parm=2 A030755 P 4 io 1 parm=3 A030756 P 4 io 1 parm=4 A030757 A 1 do 1 A030758 A 1 do 2 A030759 T 1 do 3 A030760 I 1 do 1 A030761 J 1 do 1 A030762 K 1 do 1 A030763 P 1 do 1 parm=1 A030764 P 1 do 1 parm=2 A030765 P 1 do 1 parm=3 A030766 P 1 do 1 parm=4 A030767 A 2 do 1 A030768 A 2 do 2 A030769 T 2 do 3 A030770 I 2 do 1 A030771 J 2 do 1 A030772 K 2 do 1 A030773 P 2 do 1 parm=1 A030774 P 2 do 1 parm=2 A030775 P 2 do 1 parm=3 A030776 P 2 do 1 parm=4 A030777 A 3 do 1 A030778 A 3 do 2 A030779 T 3 do 3 A030780 I 3 do 1 A030781 J 3 do 1 A030782 K 3 do 1 A030783 P 3 do 1 parm=1 A030784 P 3 do 1 parm=2 A030785 P 3 do 1 parm=3 A030786 P 3 do 1 parm=4 A030787 A 4 do 1 A030788 A 4 do 2 A030789 T 4 do 3 A030790 I 4 do 1 A030791 J 4 do 1 A030792 K 4 do 1 A030793 P 4 do 1 parm=1 A030794 P 4 do 1 parm=2 A030795 P 4 do 1 parm=3 A030796 P 4 do 1 parm=4 A051120 A 1 do 3 offset=0 first=2 A055168 B 0 fa 3 with0=2 A055169 N 0 fa 3 with0=2 A055170 D 0 fa 2 with0=2 A055171 B 1 fa 3 A055172 N 1 fa 3 A055173 D 1 fa 2 A055174 B 2 fa 3 A055175 N 2 fa 3 A055176 D 2 fa 2 A055177 B 3 fa 3 A055178 N 3 fa 3 A055179 D 3 fa 2 A055180 B 4 fa 3 A055181 N 4 fa 3 A055182 D 4 fa 2 A055183 B 5 fa 3 A055184 N 5 fa 3 A055185 D 5 fa 2 A055186 A 0 io 3 with0=3 A055187 A 1 io 3 with0=2 A055188 A 2 fa 3 A055189 A 3 fa 3 A055190 A 4 fa 3 A055191 A 5 fa 3 A079668 A 1 io 3 first=2 with0=3 A079686 A 0 do 3 offset=0 with0=3 A126027 T 1 io 5 with0=3 A174382 A 0 io 5 with0=3 A217760 A 0 fa 3 with0=2 A217780 A 1 fa 3 A240508 T 0 io 5 with0=3 122 sequences
Generating Perl program
The Perl program below can be used to generate all sequences of the list. It produces b-files, but the rules T, J, K are currently rather slow, and take time above 100 entries. The latest version of the program and its auxilliary files can be found on a Github repository.
#!perl
# Generate OEIS A030707, A055187, A217760 and related
# "cumulative counting" sequences as defined by Clark Kimberling.
# http://faculty.evansville.edu/ck6/integer/unsolved.html, Problem 4
# @(#) $Id$
# 2018-04-20, Georg Fischer (previosu version in cumulcount2.pl)
#------------------------------------------------------
# Comment from A217760:
# Write 0 followed by segments defined inductively as follows: each segment
# tells how many times each previously written integer occurs, in the order
# of first occurrence. This is Method A (adjective-before-noun pairs); for
# Method B (noun-before-adjective), see A055168.
# Example:
# Start with 0, followed by the adjective-noun pair 1,0; followed by
# adjective-noun pairs 2,0 then 1,1; etc. Writing the pairs vertically,
# the initial segments are
# 0.|.1.|.2 1.|.3 3 1.|.4 5 2 2.|.5 6 5 3 1 1.|.6 9 6 5 2 4 1.|.7 11 8 6 4 6 4 1
# ..|.0.|.0 1.|.0 1 2.|.0 1 2 3.|.0 1 2 3 4 5.|.0 1 2 3 4 5 6.|.0 1 2 3 4 5 6 9
#
# Usage:
# perl cumulcount.pl -m method -r row -n noeis -l len -a appear -o offset -s start -p parm -w with0 -d debug
# All parameters are optional and have a default value:
# method = "A" (attribute over noun; default)
# "B" (noun over attribute)
# "D" (new, distinct elements)
# "I" (inverse, first occurrence of a number)
# "J" (next term which is greater than all previous)
# "K" (next position where term is greater than all previous)
# "N" (number of new elements in segment)
# "P" (positions of small numbers (parm))
# "S" (sum of terms in segment n)
# "T" (number of terms in segment n)
# row = 3 (count in both rows, output both; default)
# 1 (count in both rows, output 1st)
# 2 (count in both rows, output 2nd)
# 5 (count in 1st row only, output 1st)
# 6 (count in 1st row only, output 2nd)
# noeis = "030707|055187|217760 ..." (OEIS number without "A", default "030707")
# len = length of sequence to be generated (default: 256)
# appear = "io" (increasing order; default)
# "do" (decreasing order)
# "fa" (order of first appearance)
# first = 0, 2 (first index in @segment)
# offset = 0, 1 (index f 1st b-file entry, default: 1)
# start = starting value for a(1): 0, 1 (default), 3, 4, 5
# parm = 2nd parameter (for rule "P"): 1, 2, 3, 4
# with0 = 0 (0 is not counted for neither attr nor noun)
# 1 (0 is counted for attr only)
# 2 (0 is counted for noun only)
# 3 (0 is counted for both attr and noun)
# debug = 0 (none; default)
# 1 (with segments)
#--------------------------------------------------------
use strict;
my $appear = "io";
my $debug = 0;
my $first = 0;
my $len = 256;
my $method = "A";
my $noeis = "030707";
my $offset = 1;
my $parm = 0;
my $row = 3; # count in both rows, output both; default
my $start = 1;
my $with0 = 0;
while (scalar(@ARGV) > 0) {
my $opt = shift(@ARGV);
if (0) {
} elsif ($opt eq "-a") { $appear = shift(@ARGV);
} elsif ($opt eq "-d") { $debug = shift(@ARGV);
} elsif ($opt eq "-f") { $first = shift(@ARGV);
} elsif ($opt eq "-l") { $len = shift(@ARGV);
} elsif ($opt eq "-m") { $method = shift(@ARGV);
} elsif ($opt eq "-n") { $noeis = shift(@ARGV);
} elsif ($opt eq "-o") { $offset = shift(@ARGV);
} elsif ($opt eq "-p") { $parm = shift(@ARGV);
} elsif ($opt eq "-r") { $row = shift(@ARGV);
} elsif ($opt eq "-s") { $start = shift(@ARGV);
} elsif ($opt eq "-w") { $with0 = shift(@ARGV);
} else { die "invalid option \"$opt\"\n";
}
} # while ARGV
if ($debug == 99) {
print " [http://oeis.org/A$noeis A$noeis] $method $start $appear $row";
if ($offset != 1) { print " offset=$offset"; }
if ($first != 0) { print " first=$first"; }
if ($parm != 0) { print " parm=$parm"; }
if ($with0 != 0) { print " with0=$with0"; }
print "\n";
exit(0);
}
my ($sec, $min, $hour, $mday, $mon, $year, $wday, $yday, $isdst)
= localtime (time);
my $timestamp = sprintf ("%04d-%02d-%02d %02d:%02d"
, $year + 1900, $mon + 1, $mday, $hour, $min);
print <<"GFis";
# http://oeis.org/A$noeis/b$noeis.txt: table n,a(n) for n=1..$len
# Generated on $timestamp by
# perl cumulcount.pl -m $method -r $row -n $noeis -l $len -a $appear -f $first -o $offset -s $start -p $parm -w $with0
GFis
my %inverse = (); # inverse sequence
my $curmax = $start - 1;
my $segno = 0;
my @segment = ();
# $segment[i+0] = attribute, how often (i = 1, 3, 5 ..)
# $segment[i+1] = noun, which number is counted,
# always this order, increasing nouns, always complete with zero attributes
my @count; # temporary copy of the attributes
my @app1st; # 1st appearance of a noun
my @seqlen = (1); # cumulative length of sequence so far, indexed with $segno
my $attr; # attribute, count of nouns
my $noun; # the numbers to be counted
my $iseg; # index in @segment
# first segment
$noun = 0;
while ($noun < $start) { # fill before $start
push(@segment, 0, $noun);
$noun ++;
} # while filling
push(@segment, 1, $start);
# first b-file entry
my $k = $offset;
my $k2 = $k; # copy of k, running as if it were rule A
if (0) {
} elsif ($method =~ m{[ABIJKP]}i) {
if (($row & 1) != 0) {
&bfile($start);
}
} elsif ($method =~ m{[D]}i) {
print "$k $start\n"; $k ++;
} elsif ($method =~ m{[N]}i) {
# print "$k 1\n" ; $k ++;
$seqlen[0] = 0;
} elsif ($method =~ m{[S]}i) {
} elsif ($method =~ m{[T]}i) {
if ($noeis eq "240508") {
print "$k 1\n"; $k ++;
}
} else {
die "invalid method \"$method\" at bf(1)\n";
}
push(@app1st, $start);
$segno ++;
# main loop
while ($k <= $len and $segno <= $len) { # compute new segment from current
&advance();
$segno ++;
} # while b-file
if ($method =~ m{I}i) { # special treatment of the inverse
$k = 1;
foreach my $attr (sort {$a <=> $b} (keys(%inverse))) {
last if $attr > $k; # must be monotone
print "$k $inverse{$attr}\n"; $k ++;
} # foreach
} # method I
#----------------
sub advance { # count between 0 and $nmax, and store in @counts
my $amax = -1; # $nmax is the current segment length / 2
if (1) { # compute length of current segment
$seqlen[$segno] = 0; # number of elements in segment
for ($iseg = $first; $iseg < scalar(@segment); $iseg += 2) { # increment for valid entries
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
if ($attr != 0 or ($with0 & 1) != 0) {
$seqlen[$segno] ++;
}
} # while incrementing
} # segment length
if ($debug >= 1) {
print "seg#$segno:";
for ($iseg = $first; $iseg < scalar(@segment); $iseg += 2) { # print the elements of this segment
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
print " $attr.$noun";
} # while copying
print " seqlen=$seqlen[$segno]\n";
print "app1st: " . join(" ", @app1st) . "\n";
} # debug
# now the b-file entries
if (0) {
} elsif ($method =~ m{[ABIJKP]}i) { # first or second row or both
if (0) {
} elsif ($appear eq "fa") { # order of first appearance
for (my $iapp = 0; $iapp < scalar(@app1st) ; $iapp ++) {
$iseg = $app1st[$iapp] << 1;
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
if ($attr != 0 or ($with0 & 1) != 0) {
&bfile($attr, $noun); # for method I: store %inverse only
}
} # for
} elsif ($appear eq "do") { # decreasing order
for ($iseg = scalar(@segment) - 2; $iseg >= $first; $iseg -= 2) {
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
if ($attr != 0 or ($with0 & 1) != 0) {
&bfile($attr, $noun); # for method I: store %inverse only
}
} # for
} elsif ($appear eq "io") { # increasing order
for ($iseg = $first; $iseg < scalar(@segment) ; $iseg += 2) {
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
if ($attr != 0 or ($with0 & 1) != 0) {
&bfile($attr, $noun); # for method I: store %inverse only
}
} # for
} else {
die "invalid parameter op=\"$appear\"\n";
}
} elsif ($method =~ m{[D]}i) { # new terms (for $appear eq "fa")
if ($debug >= 1) {
print "range " . ($seqlen[$segno - 1]) . ".." . ($seqlen[$segno] - 1) . "\n";
}
for (my $iapp = $seqlen[$segno - 1]; $iapp < $seqlen[$segno]; $iapp ++) {
$iseg = $app1st[$iapp] << 1;
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
if ($attr != 0 or ($with0 & 1) != 0) {
&bfile($attr, $noun);
}
} # for
} elsif ($method =~ m{[N]}i) { # no. of new terms in segment
&bfile($seqlen[$segno] - $seqlen[$segno - 1]);
} elsif ($method =~ m{[T]}i) { # no. of terms in segment
&bfile($seqlen[$segno]);
}
#--------
# compute following segment
for ($iseg = $first; $iseg < scalar(@segment) ; $iseg += 2) { # copy attr and determine maximum attr
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
$count[$noun] = $attr; # copy old attr
if ($attr > $amax) {
$amax = $attr;
}
} # while copying
my $last_noun = $noun;
$noun = $last_noun + 1;
while ($noun <= $amax) { # insert nouns with 0 attributes
$count[$noun ++] = 0;
} # while inserting
my $ff_count = $noun;
# now add all (or row1, row2) to @count
if (0) {
} elsif ($appear eq "fa") {
for (my $iapp = 0; $iapp < $seqlen[$segno]; $iapp ++) {
&assemble($app1st[$iapp] << 1);
} # for $iapp
} else { # "io", "do"
for ($iseg = $first; $iseg < scalar(@segment); $iseg += 2) { # add
&assemble($iseg);
} # for $iseg
if (0 and $noeis eq "055186") {
} # 055186
} # "io", "do"
# copy it back to the segment
$iseg = 0;
$noun = 0;
while ($noun < $ff_count) { # add
$segment[$iseg + 0] = $count[$noun];
$segment[$iseg + 1] = $noun;
$iseg += 2;
$noun ++;
} # while copying back
} # sub advance
#----------------
sub assemble {
my ($iseg) = @_;
$attr = $segment[$iseg + 0];
$noun = $segment[$iseg + 1];
if (($attr != 0 or ($with0 & 1) != 0) and ($row != 6)) {
if ($count[$attr] == 0) { # appears for the first time
push(@app1st, $attr);
}
$count[$attr] ++;
}
if (($attr != 0 or ($with0 & 1) != 0) and ($noun != 0 or ($with0 & 2) != 0) and ($row != 5)) {
$count[$noun] ++;
}
if ($attr == 0 and $noeis eq "079668") {
$first = 0;
}
} # assemble
#----------------
sub bfile {
if ($k > $len) {
return;
}
my ($attr, $noun) = @_;
if (0) {
} elsif ($method =~ m{P}i) {
if ($attr == $parm) {
print "$k $k2\n"; $k ++;
}
$k2 ++;
} elsif ($method =~ m{I}i) {
if (! defined($inverse{$attr})) {
# assume that method "I" is called with row=1 only !
$inverse{$attr} = $k;
if ($debug >= 1) {
print "# stored $k in inverse{$attr}\n";
}
}
$k ++;
} elsif ($method =~ m{J}i) {
if ($attr > $curmax) {
print "$k $attr\n"; $k ++;
$curmax = $attr;
}
$k2 ++;
} elsif ($method =~ m{K}i) {
if ($attr > $curmax) {
print "$k $k2\n"; $k ++;
$curmax = $attr;
}
$k2 ++;
} elsif (scalar(@_) == 1) {
print "$k $attr\n"; $k ++;
} elsif ($method =~ m{N}i) {
print "$k $attr\n"; $k ++;
} elsif ($method =~ m{T}i) {
print "$k $attr\n"; $k ++;
} elsif ($method =~ m{[AD]}i) { # attribute before noun
if (($row & 1) != 0) {
print "$k $attr\n"; $k ++;
}
if (($row & 2) != 0 and $k <= $len) {
print "$k $noun\n"; $k ++;
}
} elsif ($method =~ m{[BD]}i) { # noun before attribute
if (($row & 1) != 0) {
print "$k $noun\n"; $k ++;
}
if (($row & 2) != 0 and $k <= $len) {
print "$k $attr\n"; $k ++;
}
} else {
die "invalid method \"$method\" in sub bfile\n";
}
} # bfile
__DATA__
Rule A, A055187:
1 | 1 | 3 | 4 1 | 6 2 1 | 8 1 3 2 1 |
| 1 | 1 | 1 3 | 1 3 4 | 1 2 3 4 6 |
| 11 3 5 3 2 1 | 13 5 8 4 1 3 2 1 |
| 1 2 3 4 6 8 | 1 2 3 4 5 6 8 11 |
| 16 7 10 6 3 4 4 2 1 |
| 1 2 3 4 5 6 8 11 13 |
| 18 9 12 9 4 6 1 5 1 3 2 1 |
| 1 2 3 4 5 6 7 8 10 11 13 16 |
| 22 11 14 11 6 8 2 6 2 2 4 1 3 2 1 |
| 1 2 3 4 5 6 7 8 9 10 11 12 13 16 18 |
| 25 16 16 14 ...
| 1 2 3 4 ...
--------------------
A030717 (row 1), A030718 (row2) - zeroes are inconsistent
1 | 1 | 2 | 2, 1 | 3, 2 | 3, 3, 1 | 4, 3, 3 | 4, 3, 5, 1 | 5, 3, 6, 2, 1
| 1 | 1 | 1, 2 | 1, 2 | 1, 2, 3 | 1, 2, 3 | 1, 2, 3, 4 | 1, 2, 3, 4, 5
| 6, 4, 7, 2, 2, 1 | 7, 6, 7, 3, 2, 2, 1| 8, 8, 8, 3, 2, 3, 3
| 1, 2, 3, 4, 5, 6 | 1, 2, 3, 4, 5, 6, 7| 1, 2, 3, 4, 5, 6, 7
| 8, 9, 11, 3, 2, 3, 3, 3 | 8, 10, 15, 3, 2, 3, 3, 4, 1, 0, 1
| 1, 2, 3, 4, 5, 6, 7, 8 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11
| 10, 11, 18, 4, 2, 3, 3, 5, 1, 1, 1, 0, 0, 0, 1
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15
| 14, 12, 20, 5, 3, 3, 3, 5, 1, 2, 2, 0, 0, 0, 1, 0, 0, 1
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 18
| 17, 14, 23, 5, 5
| 1, 2, 3 ...
#----------------------------------------------
A051120 ,1|1,1|3,1|1,3,4,1|1,4,2,3,6,1|1,6,2,4,3,3,1,2,8,1|1,8,2,6,3,4,5,3,3,2,11,1|1,11,2,8,3,6,1,5,4,4,8,3,5,2,13,1|1,13,2,11,4,8,4,6,3,5,6,4,10,3,7,2,16,1,1,16,2,13,3,11,1,10,5,8,1,7,6,6,4,5,9,4,12,3,9,2,18,1,
A055187 ,1|1,1|3,1|4,1,1,3|6,1,2,3,1,4|8,1,1,2,3,3,2,4,1,6|11,1,3,2,5,3,3,4,2,6,1,8|13,1,5,2,8,3,4,4,1,5,3,6,2,8,1,11|16,1,7,2,10,3,6,4,3,5,4,6,4,8,2,11,1,13,18,1,9,2,12,3,9,4,4,5,6,6,1,7,5,8,1,10,3,11,2,13,1,16,22,1,
makefile
A Unix makefile was used
- to produce the list of sequences,
- to download (once) the corresponding b-files,
- to regenerate the particular sequences and
- to compare them against the stored b-files.
#!make # Generate OEIS A030707, A055187, A217760 and related # "cumulative counting" sequences as defined by Clark Kimberling. # http://faculty.evansville.edu/ck6/integer/unsolved.html, Problem 4 # @(#) $Id$ # 2018-04-27, Georg Fischer (previosu version in makefile2) # # C.f. listing.tmp (produced by target "listing", below) # Needs cumulcount.pl in current directory, and # Unix tools diff, echo, grep, head, make, perl, sed, tee, test, wget #--------------------------------- TARGET=cumulcount SIL= LIMIT=100 APPEAR=io DEBUG=0 FIRST=0 LEN=$(LIMIT) OFFSET=1 PARM=0 ROW=3 START=1 WITH0=0 all: 0307nn 0551nn misc quick: 030707 030708 030717 030718 055168 055186 055187 misc 0307nn: 0307io 0307do 0551nn: 05516n 05518n #---------- misc: 051120 \ 079668 079686 \ 126027 253170 \ 174382 240508\ 2177nn 051120: # A055187, but decreasing make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 OFFSET=0 FIRST=2 079668: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 WITH0=3 FIRST=2 079686: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=1000 APPEAR=do START=0 WITH0=3 OFFSET=0 126027: # row lengths of A030717; trick with OFFSET 2: 1st entry is missing make $(SIL) $(TARGET) METHOD=T ROW=5 NOEIS=$@ LEN=24 APPEAR=io START=1 WITH0=3 253170: # row sums of A030717 # nyi 174382: # like 030717 make $(SIL) $(TARGET) METHOD=A ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=0 WITH0=3 240508: # row lengths in A174382; 1st two lines wrong make $(SIL) $(TARGET) METHOD=T ROW=5 NOEIS=$@ LEN=24 APPEAR=io START=0 WITH0=3 2177_list: grep -aE "^A2177" ../names 2177nn: 217760 217780 # ok 217760: # c.f. 055186 make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=10000 APPEAR=fa START=0 WITH0=2 217780: # c.f. 055187 make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=10000 APPEAR=fa START=1 #---------- 0307_list: grep -aE "^A0307" ../names 0307_compare: diff -w -y --width=32 --suppress-common-lines b030709.txt b030759.txt || : diff -w -y --width=32 --suppress-common-lines b030729.txt b030779.txt || : diff -w -y --width=32 --suppress-common-lines b030739.txt b030769.txt || : diff -w -y --width=32 --suppress-common-lines b030749.txt b030799.txt || : diff -w -y --width=32 --suppress-common-lines b030711.txt b030761.txt || : 0307io: 03070n 03071n 03072n 03073n 03074n 03070n: 030707 030708 030709 030710 030711 030712 030713 030714 030715 030716 03071n: 030717 030718 030719 030720 030721 030722 030723 030724 030725 030726 03072n: 030727 030728 030729 030730 030731 030732 030733 030734 030735 030736 03073n: 030737 030738 030739 030740 030741 030742 030743 030744 030745 030746 03074n: 030747 030748 030749 030750 030751 030752 030753 030754 030755 030756 0307do: 03075n 03076n 03077n 03078n 03075n: 030757 030758 030759 030760 030761 030762 030763 030764 030765 030766 03076n: 030767 030768 030769 030770 030771 030772 030773 030774 030775 030776 03077n: 030777 030778 030779 030780 030781 030782 030783 030784 030785 030786 03078n: 030787 030788 030789 030790 030791 030792 030793 030794 030795 030796 030707: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030708: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030709: make $(SIL) $(TARGET) METHOD=T ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030710: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=io START=1 030711: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030712: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030713: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=1 030714: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=2 030715: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=3 030716: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=4 030717: make $(SIL) $(TARGET) METHOD=A ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 WITH0=3 030718: make $(SIL) $(TARGET) METHOD=A ROW=6 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030719: make $(SIL) $(TARGET) METHOD=T ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030720: make $(SIL) $(TARGET) METHOD=I ROW=5 NOEIS=$@ LEN=2000 APPEAR=io START=1 030721: make $(SIL) $(TARGET) METHOD=J ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030722: make $(SIL) $(TARGET) METHOD=K ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 030723: make $(SIL) $(TARGET) METHOD=P ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=1 030724: make $(SIL) $(TARGET) METHOD=P ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=2 030725: make $(SIL) $(TARGET) METHOD=P ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=3 030726: make $(SIL) $(TARGET) METHOD=P ROW=5 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 PARM=4 030727: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 030728: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 030729: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 030730: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=io START=3 030731: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 030732: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 030733: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 PARM=1 030734: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 PARM=2 030735: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 PARM=3 030736: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=3 PARM=4 030737: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 030738: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 030739: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 030740: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=io START=2 030741: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 030742: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 030743: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 PARM=1 030744: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 PARM=2 030745: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 PARM=3 030746: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=2 PARM=4 030747: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 030748: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 030749: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 030750: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=io START=4 030751: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 030752: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 030753: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 PARM=1 030754: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 PARM=2 030755: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 PARM=3 030756: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=4 PARM=4 030757: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 030758: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 030759: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 030760: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=do START=1 030761: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 030762: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 030763: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 PARM=1 030764: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 PARM=2 030765: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 PARM=3 030766: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=1 PARM=4 030767: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 030768: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 030769: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 030770: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=do START=2 030771: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 030772: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 030773: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 PARM=1 030774: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 PARM=2 030775: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 PARM=3 030776: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=2 PARM=4 030777: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 030778: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 030779: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 030780: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=do START=3 030781: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 030782: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 030783: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 PARM=1 030784: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 PARM=2 030785: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 PARM=3 030786: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=3 PARM=4 030787: make $(SIL) $(TARGET) METHOD=A ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 030788: make $(SIL) $(TARGET) METHOD=A ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 030789: make $(SIL) $(TARGET) METHOD=T ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 030790: make $(SIL) $(TARGET) METHOD=I ROW=1 NOEIS=$@ LEN=2000 APPEAR=do START=4 030791: make $(SIL) $(TARGET) METHOD=J ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 030792: make $(SIL) $(TARGET) METHOD=K ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 030793: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 PARM=1 030794: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 PARM=2 030795: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 PARM=3 030796: make $(SIL) $(TARGET) METHOD=P ROW=1 NOEIS=$@ LEN=$(LIMIT) APPEAR=do START=4 PARM=4 #---------- 0551_list: grep -aE "^A0551" ../names 05516n: 055168 055169 055170\ 055171 055172 055173\ 055174 055175 055176\ 055177 055178 055179\ 055180 055181 055182\ 055183 055184 055185 055168: make $(SIL) $(TARGET) METHOD=B ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=0 WITH0=2 055169: make $(SIL) $(TARGET) METHOD=N ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=0 WITH0=2 055170: make $(SIL) $(TARGET) METHOD=D ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=0 WITH0=2 055171: make $(SIL) $(TARGET) METHOD=B ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=1 055172: make $(SIL) $(TARGET) METHOD=N ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=1 055173: make $(SIL) $(TARGET) METHOD=D ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=1 055174: make $(SIL) $(TARGET) METHOD=B ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=2 055175: make $(SIL) $(TARGET) METHOD=N ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=2 055176: make $(SIL) $(TARGET) METHOD=D ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=2 055177: make $(SIL) $(TARGET) METHOD=B ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=3 055178: make $(SIL) $(TARGET) METHOD=N ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=3 055179: make $(SIL) $(TARGET) METHOD=D ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=3 055180: make $(SIL) $(TARGET) METHOD=B ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=4 055181: make $(SIL) $(TARGET) METHOD=N ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=4 055182: make $(SIL) $(TARGET) METHOD=D ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=4 055183: make $(SIL) $(TARGET) METHOD=B ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=5 055184: make $(SIL) $(TARGET) METHOD=N ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=5 055185: make $(SIL) $(TARGET) METHOD=D ROW=2 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=5 #----- 05518n: 055186 055187 \ 055188 055189 055190 055191 055186: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=0 WITH0=3 055187: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=io START=1 WITH0=2 055188: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=2 055189: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=3 055190: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=4 055191: make $(SIL) $(TARGET) METHOD=A ROW=3 NOEIS=$@ LEN=$(LIMIT) APPEAR=fa START=5 #--------------------------------------------------------------- cumulcount: # ------------------------------------------------------- test -s b$(NOEIS).txt || wget http://oeis.org/A$(NOEIS)/b$(NOEIS).txt grep -E "^[0-9]" b$(NOEIS).txt | head -n $(LEN) > bf.tmp # wc -l bf.tmp perl cumulcount.pl -m $(METHOD) -r $(ROW) -n $(NOEIS) -l $(LEN) -a $(APPEAR) -f $(FIRST) -o $(OFFSET) -s $(START) -p $(PARM) -w $(WITH0) -d $(DEBUG) > A$(NOEIS).pn.tmp grep -E "^[0-9]" A$(NOEIS).pn.tmp | head -n `wc -l bf.tmp | cut -d " " -f 1` > pn.tmp # diff -w -y --width=32 bf.tmp pn.tmp || : diff -w -y --suppress-common-lines --width=32 bf.tmp pn.tmp | head -n 8 # debug: # ------------------------------------------------------- perl cumulcount.pl -m $(METHOD) -r $(ROW) -n $(NOEIS) -l $(LEN) -a $(APPEAR) -f $(FIRST) -o $(OFFSET) -s $(START) -p $(PARM) -w $(WITH0) -d 1 | less #-------- list1: perl cumulcount.pl -m $(METHOD) -r $(ROW) -n $(NOEIS) -l $(LEN) -a $(APPEAR) -f $(FIRST) -o $(OFFSET) -s $(START) -p $(PARM) -w $(WITH0) -d 99 # echo " " [https://oeis.org/A$(NOEIS) A$(NOEIS)] Rule $(RULE).$(ROW) $(ORD) $(P1) $(P2) " "\ # `wc -l b$(NOEIS).txt | cut -d " " -f 1` listing: make -s all TARGET=list1 | sort \ | tee listing.tmp wc -l listing.tmp #-------- # e.g. https://oeis.org/search?q=id:A257908&fmt=text intern1: test -s A$(NOEIS).int || wget -O A$(NOEIS).int "http://oeis.org/search?q=id:A$(NOEIS)\&fmt=text" internal: make $(SIL) all TARGET=intern1