OEIS/Negative-Positive

From tehowiki
Revision as of 14:59, 26 March 2018 by imported>Gfis (results)
Jump to navigation Jump to search

Clark Kimberling defined - for example - A131388 with the following rules: 

Rule 1 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
Step 2:  Let h be the least positive integer not in D(k) such that a(k) + h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
Conjecture:  if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0).  Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.

Depending on the starting values a(1) and d(1), and with variations of the rules, there are several dozens of related sequences in the OEIS. There are lists of related sequences in A257705, A257883 and A257905.

List of Negative-Positive sequences

The list below tries to combine the partial lists of such sequences, and to describe the variations. The list has the following entries:

  • the OEIS A-number,
  • the major rule number as defined by C. Kimberling, except for "Rule 4" which he calls the "Algorithm",
  • a minor rule number which gives the first constant (s = 0, 1) in Step 1 above: "... such that s - a(k) < h < 0 ...",
  • a letter code for the type of the sequence:
    • a for a(k),
    • d for d(k),
    • in for inverse(a(k)),
    • cp for positions of positive integers,
    • cn for positions of negative integers,
    • d0 for positions of non-negative terms,
    • dn for positions of negative terms,
  • the starting values a(1) and d(1).
A131388 Rule 1.1 a  1 0
A131389 Rule 1.1 d  1 0
A131390 Rule 1.1 in 1 0
A131391 Rule 1.1 cp 1 0
A131392 Rule 1.1 cn 1 0
A131393 Rule 2.1 a  1 0
A131394 Rule 2.1 d  1 0
A131395 Rule 2.1 in 1 0
A131396 Rule 2.1 cp 1 0
A131397 Rule 2.1 cn 1 0
A175007 Rule 1.1 d0 1 0
A175008 Rule 1.1 dn 1 0
A175498 Rule 4.0 a  1 0
A175499 Rule 4.0 d  0 1
A257705 Rule 1.1 a  0 0
A257706 Rule 1.0 a  0 1
A257876 Rule 1.1 a  0 2
A257877 Rule 1.1 a  0 3
A257878 Rule 1.1 a  1 1
A257879 Rule 1.0 a  2 0
A257880 Rule 1.0 d  2 0
A257881 Rule 1.0 a  2 1
A257882 Rule 1.0 a  2 2
A257883 Rule 4.0 a  0 0
A257884 Rule 4.0 a  0 1
A257885 Rule 4.0 a  0 2
A257902 Rule 4.0 d  0 2
A257903 Rule 4.0 a  0 3
A257904 Rule 4.0 d  0 3
A257905 Rule 3.1 a  0 0
A257906 Rule 3.1 a  0 1
A257907 Rule 3.1 d  0 1
A257908 Rule 3.0 a  0 2
A257909 Rule 3.0 d  0 2
A257910 Rule 3.0 a  0 3
A257911 Rule 4.0 a  2 2
A257912 Rule 4.0 d  2 2
A257915 Rule 1.1 d  0 3
A257918 Rule 1.0 d  2 2
A257980 Rule 3.1 d  0 3
A257981 Rule 3.1 a  1 1
A257982 Rule 3.1 d  1 1
A257983 Rule 3.1 a  1 2
A257985 Rule 3.1 a  2 0
A257986 Rule 3.1 a  2 1
A257987 Rule 3.1 a  2 2
A258046 Rule 3.1 a  1 0
A258047 Rule 3.1 d  1 0

Generating Perl program

The Perl program below can be used to generate all sequences of the list. It produces b-files with up to 100,000 terms rather rather quickly. The program takes parameters as described in the header. It contains the lost code portion of Rule 2 mentioned in the comment for A131393, and it implements the different variants depending on the parameter $op. 


#!perl

# Generate OEIS Sequence A131393 and its companions
# as defined by Clark Kimberling
# @(#) $Id$
# 2018-02-24, Georg Fischer
#------------------------------------------------------
# C.f. list of sequences in https://oeis.org/search?q=A257705
# usage:
#   perl negpos.pl rule s noeis n op a1 d1
#       rule  = 1|2|3|4
#       s     = 0|1
#       noeis = "131388|131389|131393|131394..." (without "A")
#       n     = length of sequence to be generated
#       op    = ak, dk, cp, cn, dp(positive d(K)), dn(negative d(k)), in(inverse) 
#       a1    = starting value for a(1)
#       d1    = starting value for d(1)
#------------------------------------------------------
# Formula (Rule 1):
# a(k) = a(k-1) + d(k) 
# d(k) = max({s, s-1 ... 1-a(k-1)}) such that
#   d(k) not in d(1..k-1) and
#   a(k) not in a(1..k-1)
# if no such d(k) exists, then
# d(k) = min({1,2, ... a(k-1)}) such that
#   d(k) not in d(1..k-1) and
#   a(k) not in a(1..k-1)
# s = -1 for A131388, and s = min(-1, d(k-1)) for A131393
#--------------------------------------------------------
use strict;

my $rule = 1;   if (scalar(@ARGV) > 0) { $rule  = shift(@ARGV); }
my $s = 1;      if (scalar(@ARGV) > 0) { $s     = shift(@ARGV); }
my $noeis = ""; if (scalar(@ARGV) > 0) { $noeis = shift(@ARGV); }
my $n = 1000;   if (scalar(@ARGV) > 0) { $n     = shift(@ARGV); }
my $op = "ak";  if (scalar(@ARGV) > 0) { $op    = shift(@ARGV); }
my $a1 = 1;     if (scalar(@ARGV) > 0) { $a1    = shift(@ARGV); }
my $d1 = 0;     if (scalar(@ARGV) > 0) { $d1    = shift(@ARGV); }
my $k = 1;
my $ak = $a1; my $akm1 = $ak; my %aset = ($ak, $k);
my $dk = $d1; my $dkm1 = $dk; my %dset = ($dk, $k); # $dk is h
print "# http://oeis.org/A$noeis/b$noeis.txt:"
        . " table n,a(n),n=1..$n\n";
#    print "# ak = $ak, dk = $dk, akm1 = $akm1, dkm1 = $dkm1 \n";
    if (0) {
    } elsif ($op eq "ak") {
        print "$k $ak\n";
    } elsif ($op eq "dk") {
        print "$k $dk\n";
    }
my $busy;
$k ++;
while ($k <= $n) {
    $busy = 1;
    if (0) {
    } elsif ($rule == 1 or $rule == 2) { # for A131388, A257705 et al.
        $dk = -1; # start downwards
        if ($rule == 2 and $dkm1 < 0) { # for A131393 et al.
            $dk = $dkm1 - 1;
        }
        while ($busy == 1 and $dk > $s - $akm1) { # downwards
            $ak = $akm1 + $dk;
            if (!defined($aset{$ak}) and !defined($dset{$dk} and $ak>0)) {
                $busy=0; $aset{$ak} = $k;         $dset{$dk}=$k;
            } else {
                $dk --;
            }
        } # while downwards
        if ($busy == 1) {
            $dk = +1; # start upwards
        }
        while ($busy == 1                     ) { # upwards
            $ak = $akm1 + $dk;
            if (!defined($aset{$ak}) and !defined($dset{$dk}          )) {
                $busy=0; $aset{$ak} = $k;         $dset{$dk}=$k;
            } else {
                $dk ++;
            }
        } # while upwards

    } elsif ($rule == 3) { # for A257905, 908
    	# print "$k $akm1 dk=$dkm1\n";
        $dk = $s - $akm1 + 1; # start upwards in negative
        while ($busy == 1 and $dk < 0) { 
            $ak = $akm1 + $dk;
            if (!defined($aset{$ak}) and !defined($dset{$dk} and $ak>0)) {
                $busy=0; $aset{$ak} = $k;         $dset{$dk}=$k;
            } else {
                $dk ++;
            }
        } # while negative
        if ($busy == 1) {
            $dk = +1; # start upwards
        }
        while ($busy == 1                     ) { # upwards
            $ak = $akm1 + $dk;
            if (!defined($aset{$akm1 - $dk}) and !defined($dset{$dk}          )) {
                $busy=0; $aset{$ak        } = $k;         $dset{$dk}=$k;
            } else {
                $dk ++;
            }
        } # while upwards

    } elsif ($rule == 4) { # "Algorithm" for A257883 et al.
        $dk = $s - $ak + 1;
        while ($busy == 1                      ) { # upwards
            $ak = $akm1 + $dk;
            if (!defined($aset{$ak}) and !defined($dset{$dk} and $ak>0)) {
                $busy=0; $aset{$ak} = $k;         $dset{$dk}=$k;
            } else {
                $dk ++;
            }
        } # while upwards
    }
    if (0) {
    } elsif ($op eq "ak") {
        print "$k $ak\n";
    } elsif ($op eq "dk") {
        print "$k $dk\n";
    }
    # print "\t# ak = $ak, dk = $dk, akm1 = $akm1, dkm1 = $dkm1\n";
    $akm1 = $ak; $dkm1 = $dk;
    $k ++; # iterate
} # while $k
#--------
if ($op !~ m{ak|dk}) { # output of operations other than "ak", "dk"
    my @ainv = sort(map { $_ = sprintf("%06d %d", $_, $aset{$_}); $_ } keys(%aset));
    my @dpos = sort(map { $_ = sprintf("%06d %d", $dset{$_}, $_); $_ } keys(%dset));
    # my $temp = shift(@dpos); # accounts for positions "-1"
    # print join("\n", @dpos) . "\n";
    if (0) {
    } elsif ($op =~ m{in}) {
        $k = 0;
        $busy = 1;
        while ($busy == 1 and $k < scalar(@ainv)) {
            my ($j, $aj) = split(/ /, $ainv[$k]);
            $j += 0; # removes the leading zeroes
            $busy = ($j == $k + 1 ? 1 : 0);
            if ($busy == 1) {
                print "$j $aj\n";
            }
            $k ++;
        } # while $k
    } elsif ($op =~ m{cp|dp|c0|d0}) {
        my $k = 0;
        print join("", map { my ($j, $dj) = split(/\s+/); 
                $j = ($op =~ m{\Ac}) ? $j - 1 : $j + 0; 
                $_ = "";
                if ($dj > (($op =~ m{0}) ? -1 : 0)) {
                    $k ++;
                    $_ = "$k $j\n";
                }
                $_ } @dpos) . "\n";
    } elsif ($op =~ m{cn|dn}) {
        my $k = 0;
        print join("", map { my ($j, $dj) = split(/\s+/); 
                $j = ($op =~ m{\Ac}) ? $j - 1 : $j + 0; 
                $_ = "";
                if ($dj < 0) {
                    $k ++;
                    $_ = "$k $j\n";
                }
                $_ } @dpos) . "\n";
    }
} # other oper
# https://oeis.org/wiki/User:Georg_Fischer Feb. 24, 2018

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.

Results

The trailing terms of the 2 inverse sequences A131390 and A131395 were found to be wrong, maybe because too few terms of the normal sequence were used. For A131393 and its dependants, more terms up to 1000 were computed. All other b-files showed no differences. No differences were found either during a manual check of about 1/3 of the Mathematica programs. 


#!make

# Check OEIS sequences defined by Clark Kimberling
# @(#) $Id$
# 2018-02-26: RULE=
# 2018-02-23, Georg Fischer
# 
# c.f. listing.tmp (produced by target "listing", below)
#---------------------------------
TARGET=negpos
LIMIT=2000

all: rule1 rule2 rule3 rule4
#----------
# Rule 1: Step 1 downwards, Step 2 upwards
rule1: np131388 np257705 np257879 np257905
np131388: 131388
131388: 131389 175007 175008 131390 131391 131392 
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=0
131389: 
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=1 D1=0
175007:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=d0 A1=1 D1=0
175008:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dn A1=1 D1=0
131390:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=in A1=1 D1=0
	# last 11 lines of b-file are wrong
131391:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=cp A1=1 D1=0
131392:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=cn A1=1 D1=0
#--------
# from A257705:
# a(1)  d(1)      (a(n))             (d(n))
# 0       0      A257705      A131389 except for initial terms
# 0       1      A257706      A131389 except for initial terms
# 0       2      A257876      A131389 except for initial terms
# 0       3      A257877      A257915
# 1       0      A131388      A131389
# 1       1      A257878      A131389 except for initial terms
# 2       0      A257879      A257880
# 2       1      A257881      A257880 except for initial terms
# 2       2      A257882      A257918
# Rule 1: Step 1 downwards, Step 2 upwards
np257705: 257705 257876 257877 257915 257878
257705:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=0
257876:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=2
257877:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=3
257915:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=3
257878:
	make $(TARGET) RULE=1 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=1
#--------
np257879: 257706 257879 257880 257881 257882 257918
# wrong descriptions with S=1 instead of 0
257706:
	make $(TARGET) RULE=1 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=1
257879:
	make $(TARGET) RULE=1 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=0
257880:
	make $(TARGET) RULE=1 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=2 D1=0
257881:
	make $(TARGET) RULE=1 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=1
257882:
	make $(TARGET) RULE=1 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=2
257918:
	make $(TARGET) RULE=1 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=2 D1=2
#----------------------------------------------------------------
rule2: np131393
# Rule 2 like Rule 1 with different start of $dk
np131393: 131393 131394 131395 131396 131397
131393:
	make $(TARGET) RULE=2 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=0
131394: 
	make $(TARGET) RULE=2 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=1 D1=0
131395:
	make $(TARGET) RULE=2 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=in A1=1 D1=0
	# last 8 lines of b-file are wrong
131396:
	make $(TARGET) RULE=2 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=cp A1=1 D1=0
131397:
	make $(TARGET) RULE=2 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=cn A1=1 D1=0
#----------------------------------------------------------------
# Rule 3: 
rule3:
	# echo Rule 3 
#----------------------------------------------------------------
# Rule 4: upwards from lowest
rule4: np257883
np257883: 257883 257884 257885 257903 175498 257905 175499 257908 257909 257910 257911  
257883:
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=0
# A175499 except for first initial terms

257884: 175499
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=1
175499:
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=1

257885: 257902
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=2
257902:
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=2

257903: 257904
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=3
257904:
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=3

175498:
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=0
# A175499 except for first term 

257908: 257909
	make $(TARGET) RULE=3 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=2
257909:
	make $(TARGET) RULE=3 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=2

257910: 
	make $(TARGET) RULE=3 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=3
# A257909 except for initial terms

257911: 257912
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=2
257912:
	make $(TARGET) RULE=4 S=0 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=2 D1=2

# from A257883:
# a(1)   d(1)    (a(n))       (d(n))
# 0      0      A257883      A175499 except for initial terms
# 0      1      A257884      A175499
# 0      2      A257885      A257902
# 0      3      A257903      A257904
# 1      0      A175498      A175499 except for first term
# 1      1      A257905      A175499
# 2      0      A257908      A257909
# 2      1      A257910      A257909 except for initial terms
# 2      2      A257911      A257912
#--------
np257905: \
	257905      258047 \
	257906      257907 \
	257908      257909 \
	257910      257980 \
	258046      258047 \
	257981      257982 \
	257983      257909 \
	257985      258047 \
	257986      257982 \
	257987      257909
257905: 258047
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=0
258047:
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=1 D1=0

257906: 257907
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=0 D1=1
257907:
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=1

257980:
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=0 D1=3

258046: 258047
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=0

257981: 257982
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=1
257982:
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=dk A1=1 D1=1

257983: 
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=1 D1=2
# 257909.112:

257985: 
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=0
# 258047

257986: 257982
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=1
# 257982.121:

257987:
	make $(TARGET) RULE=3 S=1 NOEIS=$@ LEN=$(LIMIT) OPER=ak A1=2 D1=2
# 257909.122:

# from A257905:
# a(1)  d(1)      (a(n))       (d(n))
# 0       0      A257905      A258047
# 0       1      A257906      A257907
# 0       2      A257908      A257909
# 0       3      A257910      A257980
# 1       0      A258046      A258047
# 1       1      A257981      A257982
# 1       2      A257983      A257909
# 2       0      A257985      A257047
# 2       1      A257986      A257982
# 2       2      A257987      A257909
#---------------------------------------------------------------
negpos:
	# -------------------------------------------------------
	echo perl negpos.pl $(RULE) $(S) $(NOEIS) $(LEN) $(OPER) $(A1) $(D1)
	test -s b$(NOEIS).txt || wget http://oeis.org/A$(NOEIS)/b$(NOEIS).txt
	perl negpos.pl $(RULE) $(S) $(NOEIS) $(LEN) $(OPER) $(A1) $(D1) > A$(NOEIS).pn.tmp
	grep -Ev "^ *#" b$(NOEIS).txt | head -n $(LIMIT) > bf.tmp 
	grep -Ev "^ *#" A$(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 || :
#--------
# chain, c.f. http://faculty.evansville.edu/ck6/integer/unsolved.html, Problem 13
chain389:
	perl a1313.pl A131389 100000 > t32k.tmp
	perl chain389.pl t32k.tmp
	# no output => no counterexample up to 100000
chain394:
	perl a1313.pl A131394 10000  > u32k.tmp
	perl chain389.pl u32k.tmp
	# no output => no counterexample up to 10000
block389:
	perl block2.pl t32k.tmp
block394:
	perl block2.pl u32k.tmp
#--------
wgetall:
	wget http://oeis.org/A131388/b131388.txt
	wget http://oeis.org/A131389/b131389.txt
	wget http://oeis.org/A131393/b131393.txt
	wget http://oeis.org/A131394/b131394.txt
#-------------------------------------------
# get the list of relevant sequence numbers
nplists: Anoeis npname npseqs
Anoeis:
	grep -E "^[0-9]" makefile | sed -e "s/^/A/" \
	| cut -b 1-7 | sort | uniq | tee $@.tmp
	wc $@.tmp
#----
# prepare the descriptions
npname: Anoeis
	grep -f Anoeis.tmp ../names    | sort \
	| sed -e "s/generated by/ /" -e "s/(in Comments) with//" \
	 -e "s/Sequence//i" \
	 -e "s/Conjectured permutation of the //" \
	 -e "s/conjectured permutation//" \
	| cut -b 1-80 \
	| tee $@.tmp
	wc $@.tmp
#----
# compare the sequence lists to the b-files
npseqs: Anoeis wgetrest
	grep -f $@.tmp ../stripped | sort | tee npseqs.tmp
	perl npseqs.pl < npseqs.tmp
#----
wgetrest: Anoeis
	cut -b 2-7 Anoeis.tmp > noeis.tmp
	cat noeis.tmp | xargs -l -i{} make wget1 NOEIS={}
wget1:
	test -s b$(NOEIS).txt || wget http://oeis.org/A$(NOEIS)/b$(NOEIS).txt
#----
listing:
	make -s all TARGET=list1 | sort \
	| sed -e "s/ ak / a  /" -e "s/ dk / d  /"\
	| tee listing.tmp
	wc listing.tmp
list1: 
	echo [https://oeis.org/A$(NOEIS) A$(NOEIS)]   Rule $(RULE).$(S)   $(OPER)   $(A1) $(D1)
#----
# e.g. https://oeis.org/search?q=id:A257908&fmt=text
internal: Anoeis
	cat Anoeis.tmp | xargs -l -i{} make intern1 ANOEIS={}
intern1:
	test -s $(ANOEIS).int || \
	wget -O $(ANOEIS).int "http://oeis.org/search?q=id:$(ANOEIS)\&fmt=text"
Retrieved from "http://teherba.org/tehowiki/index.php?title=OEIS/Negative-Positive&oldid=392"