imported>Gfis: Results at the end

2018-03-26T16:42:08Z

Results at the end

← Older revision		Revision as of 16:42, 26 March 2018
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	* to regenerate the particular sequences and		* to regenerate the particular sequences and
	* to compare them against the stored b-files.		* to compare them against the stored b-files.
	~~== Results==~~
	The trailing terms of the 2 inverse sequences [https://oeis.org/A131390 A131390] and [https://oeis.org/A131395 A131395] were found to be wrong, maybe because too few terms of the normal sequence were used. For [https://oeis.org/A131393 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.
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			== Results==
			The trailing terms of the 2 inverse sequences [https://oeis.org/A131390 A131390] and [https://oeis.org/A131395 A131395] were found to be wrong, maybe because too few terms of the normal sequence were used. For [https://oeis.org/A131393 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.

imported>Gfis: github

2018-03-26T15:06:23Z

github

← Older revision		Revision as of 15:06, 26 March 2018
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	[https://oeis.org/wiki/User:Clark_Kimberling Clark Kimberling] defined - for example - [https://OEIS.org/A131388 A131388] with the following ~~rules~~:		[https://oeis.org/wiki/User:Clark_Kimberling Clark Kimberling] defined - for example - [https://OEIS.org/A131388 A131388] with the following rule:
	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).		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 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.
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	[https://oeis.org/A258047 A258047] Rule 3.1 d 1 0		[https://oeis.org/A258047 A258047] Rule 3.1 d 1 0
	==Generating Perl program==		==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 [https://oeis.org/A131393 A131393], and it implements the different variants depending on the parameter <code>$op.</code>		The Perl program below can be used to generate all sequences of the list. It produces b-files with up to 100,000 terms rather rather quickly. The program takes parameters as described in the header. It contains the lost code portion of Rule 2 mentioned in the comment for [https://oeis.org/A131393 A131393], and it implements the different variants depending on the parameter <code>$op</code>. The latest version of the program and its auxilliary files can be found on a [https://github.com/gfis/fasces/tree/master/oeis/negpos Github repository].
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2018-03-26T14:59:18Z

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2018-03-26T12:37:16Z

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[https://oeis.org/wiki/User:Clark_Kimberling Clark Kimberling] defined - for example - [https://OEIS.org/A131388 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. [https://oeis.org/A257705 A257705] lists 15 of them.

==List of Negative-Positive sequences==
The list below gives a more extended overview. 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",
* the 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:
** <code>a</code> for a(k),
** <code>d</code> for d(k),
** <code>in</code> for inverse(a(k)),
** <code>cp</code> for positions of positive integers,
** <code>cn</code> for positions of negative integers,
** <code>d0</code> for positions of non-negative terms,
** <code>dn</code> for positions of negative terms,
* the starting values a(1) and d(1).
[https://oeis.org/A131388 A131388] Rule 1.1 a 1 0
[https://oeis.org/A131389 A131389] Rule 1.1 d 1 0
[https://oeis.org/A131390 A131390] Rule 1.1 in 1 0
[https://oeis.org/A131391 A131391] Rule 1.1 cp 1 0
[https://oeis.org/A131392 A131392] Rule 1.1 cn 1 0
[https://oeis.org/A131393 A131393] Rule 2.1 a 1 0
[https://oeis.org/A131394 A131394] Rule 2.1 d 1 0
[https://oeis.org/A131395 A131395] Rule 2.1 in 1 0
[https://oeis.org/A131396 A131396] Rule 2.1 cp 1 0
[https://oeis.org/A131397 A131397] Rule 2.1 cn 1 0
[https://oeis.org/A175007 A175007] Rule 1.1 d0 1 0
[https://oeis.org/A175008 A175008] Rule 1.1 dn 1 0
[https://oeis.org/A175498 A175498] Rule 4.0 a 1 0
[https://oeis.org/A175499 A175499] Rule 4.0 d 0 1
[https://oeis.org/A257705 A257705] Rule 1.1 a 0 0
[https://oeis.org/A257706 A257706] Rule 1.0 a 0 1
[https://oeis.org/A257876 A257876] Rule 1.1 a 0 2
[https://oeis.org/A257877 A257877] Rule 1.1 a 0 3
[https://oeis.org/A257878 A257878] Rule 1.1 a 1 1
[https://oeis.org/A257879 A257879] Rule 1.0 a 2 0
[https://oeis.org/A257880 A257880] Rule 1.0 d 2 0
[https://oeis.org/A257881 A257881] Rule 1.0 a 2 1
[https://oeis.org/A257882 A257882] Rule 1.0 a 2 2
[https://oeis.org/A257883 A257883] Rule 4.0 a 0 0
[https://oeis.org/A257884 A257884] Rule 4.0 a 0 1
[https://oeis.org/A257885 A257885] Rule 4.0 a 0 2
[https://oeis.org/A257902 A257902] Rule 4.0 d 0 2
[https://oeis.org/A257903 A257903] Rule 4.0 a 0 3
[https://oeis.org/A257904 A257904] Rule 4.0 d 0 3
[https://oeis.org/A257905 A257905] Rule 3.1 a 0 0
[https://oeis.org/A257906 A257906] Rule 3.1 a 0 1
[https://oeis.org/A257907 A257907] Rule 3.1 d 0 1
[https://oeis.org/A257908 A257908] Rule 3.0 a 0 2
[https://oeis.org/A257909 A257909] Rule 3.0 d 0 2
[https://oeis.org/A257910 A257910] Rule 3.0 a 0 3
[https://oeis.org/A257911 A257911] Rule 4.0 a 2 2
[https://oeis.org/A257912 A257912] Rule 4.0 d 2 2
[https://oeis.org/A257915 A257915] Rule 1.1 d 0 3
[https://oeis.org/A257918 A257918] Rule 1.0 d 2 2
[https://oeis.org/A257980 A257980] Rule 3.1 d 0 3
[https://oeis.org/A257981 A257981] Rule 3.1 a 1 1
[https://oeis.org/A257982 A257982] Rule 3.1 d 1 1
[https://oeis.org/A257983 A257983] Rule 3.1 a 1 2
[https://oeis.org/A257985 A257985] Rule 3.1 a 2 0
[https://oeis.org/A257986 A257986] Rule 3.1 a 2 1
[https://oeis.org/A257987 A257987] Rule 3.1 a 2 2
[https://oeis.org/A258046 A258046] Rule 3.1 a 1 0
[https://oeis.org/A258047 A258047] Rule 3.1 d 1 0
==Generating Perl program==
All the sequences in the list above can be generated quickly up to 100,000 terms with the following Perl program.
<pre>
<nowiki>
#!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
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OEIS/Negative-Positive - Revision history

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