John P. Linderman - cp's OEIS frontend

A171001 Binomial(n-k,k)^2 where k = ceiling(n/4).

1, 0, 1, 4, 9, 9, 36, 100, 225, 400, 1225, 3136, 7056, 15876, 44100, 108900, 245025, 627264, 1656369, 4008004, 9018009, 25050025, 64128064, 153165376, 344622096, 1012766976, 2538950544, 6009350400, 13521038400, 41408180100, 102252852900, 240407818596

Offset: 0

John P. Linderman, Aug 31 2010

nonn

A lower bound for A094837. Cf. A171002, A171003, A171006.

A171002 Binomial(n-k,k) * binomial(n-k-1,k+1) where k = ceiling(n/4).

Original entry on oeis.org

0, 0, 0, 3, 0, 6, 40, 150, 100, 525, 1960, 5880, 7056, 26460, 83160, 228690, 365904, 1189188, 3435432, 9018009, 17177160, 51531480, 141581440, 361032672, 773641440, 2204878104, 5859116640, 14647791600, 34178180400, 93989996100, 243891989880, 601019546490

Offset: 1

John P. Linderman, Sep 01 2010

nonn

Cf. A171001, A171003, A171006.

A171006 Binomial(n-k-1,k) * binomial(n-k,k+1) where k = ceiling(n/4).

Original entry on oeis.org

0, 0, 0, 1, 6, 1, 12, 60, 200, 150, 700, 2450, 7056, 8820, 31752, 97020, 261360, 426888, 1359072, 3864861, 10020010, 19324305, 57257200, 155739584, 393853824, 851005584, 2405321568, 6347376360, 15774544800, 37026362100, 101219995800, 261312846300

Offset: 0

John P. Linderman, Sep 02 2010

nonn

Another lower bound for A094837. Cf. A171001, A171002, A171003.

A171003 A lower bound for A094837.

Original entry on oeis.org

1, 2, 4, 9, 18, 40, 100, 225, 525, 1225, 3136, 7056, 17640, 44100, 108900, 261360, 637065, 1656369, 4008004, 10020010, 25050025, 64128064, 155739584, 393853824, 1012766976, 2538950544, 6347376360, 15774544800, 41408180100

Offset: 1

John P. Linderman, Sep 01 2010

nonn

See Perl script for explanation.

John P. Linderman, Table of n, a(n) for n = 1..50
John P. Linderman, Perl script for a lower bound on A094837

A135550 Number of bases b, 1 < b < n-1, in which n is a palindrome, allowing leading zeros when testing if a number is a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 3, 2, 4, 0, 5, 1, 3, 3, 5, 2, 6, 0, 6, 4, 2, 1, 8, 2, 4, 4, 6, 1, 8, 2, 6, 3, 4, 2, 10, 1, 3, 3, 9, 1, 8, 1, 4, 5, 4, 0, 11, 2, 6, 4, 6, 0, 8, 4, 8, 4, 2, 1, 14, 1, 4, 6, 8, 5, 7, 2, 7, 3, 6, 1, 14, 2, 3, 5, 4, 2, 9, 0, 12, 5, 4, 1, 14, 5, 3, 2, 7, 1, 13, 4, 6, 4, 2, 2

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11). So here we assume 1 < b < n-1.

Here 4 = 100 counts as a palindrome in base 2, since 00100 is palindromic.

Paul Tek, Table of n, a(n) for n = 0..10000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: LEADING0S=1 palin.pl]

Cf. A135549, A135551, A016038.

A135551 Number of bases b, 1 < b < n, in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 1, 3, 4, 2, 2, 4, 2, 4, 3, 4, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3, 4, 4, 1, 5, 2, 4, 4, 5, 1, 4, 4, 4, 4, 2, 2, 6, 2, 3, 5, 4, 5, 4, 3, 4, 2, 4, 2, 6, 3, 3, 3, 3, 2, 6, 1, 7, 3, 4, 2, 6, 5, 3, 2, 5, 2, 5, 4, 5, 4, 2, 2, 6, 2, 5, 4, 7, 2, 4, 1, 6

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11).
First occurrence in A037183.
a(n) is always less than A001221(n) except for 2 and 6; a(n) is always less than A001222(n) except for even powers of twos and 6, 12, 81, 243, 625, 729, 2187, 19683, 59049, ..., . - Robert G. Wilson v, Jul 17 2016

Robert G. Wilson v, Table of n, a(n) for n = 0..100000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: BASEDELTA=0 palin.pl]

Cf. A037183, A135549, A135550, A016038.

Essentially the same as A065531.

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 1}]]; Array[ Length@ palindromicBases@# &, 105, 0] (* Robert G. Wilson v, Oct 15 2014 *)
palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]];
f[n_] := Block[{s = Ceiling@ Sqrt@ n, b = 2, c = If[ IntegerQ@ Sqrt[4n + 1], -1, 0]}, While[b < s, If[ palQ[n, b], c++]; b++]; c + Count[ Mod[n, Range[s - 1]], 0]]; f[0] = 0; Array[f, 105, 0] (* much faster for large Ns *) (* Robert G. Wilson v, Oct 20 2014 *)

a(n) = A135549(n) + 1 for n>2; otherwise a(n) = A135549(n) = 0. - Michel Marcus, Oct 15 2014

a(n) = A126071(n) - 1. - Michel Marcus, Mar 07 2015

A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2, 2, 2, 0, 2, 3, 1, 1, 3, 1, 3, 2, 3, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 2, 3, 3, 0, 4, 1, 3, 3, 4, 0, 3, 3, 3, 3, 1, 1, 5, 1, 2, 4, 3, 4, 3, 2, 3, 1, 3, 1, 5, 2, 2, 2, 2, 1, 5, 0, 6, 2, 3, 1, 5, 4, 2, 1, 4, 1, 4, 3, 4, 3, 1, 1, 5, 1, 4, 3, 6, 1, 3, 0, 5

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11). So here we assume 1 < b < n-1.

Records for a(n)>=1 are in A107129. - Dmitry Kamenetsky, Oct 22 2015

T. D. Noe, Table of n, a(n) for n = 0..10000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: palin.pl]

Cf. A016038, A037183, A065531, A107129, A135550, A135551.

Cf. A016038 (non-palindromic numbers in any base 1 < b < n-1)

a = {0, 0, 0}; For[n = 4, n < 100, n++, c = 0; For[b = 2, b < n - 1, b++, If[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]], c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Feb 27 2008 *)
Table[cnt=0; Do[d=IntegerDigits[n,b]; If[d==Reverse[d], cnt++ ], {b,2,n-2}]; cnt, {n,0,100}] (* T. D. Noe, Feb 28 2008 *)
Table[Total[Boole[Table[PalindromeQ[IntegerDigits[n,b]],{b,2,n-2}]]],{n,0,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 14 2020 *)

a(n) = A065531(n)-1 = A126071(n)-2 for n>2. - T. D. Noe, Feb 28 2008

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User: John P. Linderman

A171001 Binomial(n-k,k)^2 where k = ceiling(n/4).

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A171002 Binomial(n-k,k) * binomial(n-k-1,k+1) where k = ceiling(n/4).

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A171006 Binomial(n-k-1,k) * binomial(n-k,k+1) where k = ceiling(n/4).

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A171003 A lower bound for A094837.

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A135550 Number of bases b, 1 < b < n-1, in which n is a palindrome, allowing leading zeros when testing if a number is a palindrome.

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A135551 Number of bases b, 1 < b < n, in which n is a palindrome.

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A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

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