A056481 -id:A056481 - cp's OEIS frontend

A284826 Irregular triangle T(n,k) for 1 <= k <= (n+1)/2: T(n,k) = number of primitive (aperiodic) palindromic structures of length n using exactly k different symbols.

1, 0, 0, 1, 0, 1, 0, 3, 1, 0, 2, 1, 0, 7, 6, 1, 0, 6, 6, 1, 0, 14, 25, 10, 1, 0, 12, 24, 10, 1, 0, 31, 90, 65, 15, 1, 0, 27, 89, 65, 15, 1, 0, 63, 301, 350, 140, 21, 1, 0, 56, 295, 349, 140, 21, 1, 0, 123, 965, 1701, 1050, 266, 28, 1

Offset: 1

Andrew Howroyd, Apr 03 2017

Permuting the symbols will not change the structure.

			Triangle starts:
1
0
0   1
0   1
0   3    1
0   2    1
0   7    6     1
0   6    6     1
0  14   25    10     1
0  12   24    10     1
0  31   90    65    15     1
0  27   89    65    15     1
0  63  301   350   140    21    1
0  56  295   349   140    21    1
0 123  965  1701  1050   266   28   1
0 120  960  1700  1050   266   28   1
0 255 3025  7770  6951  2646  462  36  1
0 238 2999  7760  6950  2646  462  36  1
0 511 9330 34105 42525 22827 5880 750 45 1
0 495 9305 34095 42524 22827 5880 750 45 1
--------------------------------------------
For n=5, structures with 2 symbols are aabaa, ababa and abbba, so T(5,2) = 3.
For n=6, structures with 2 symbols are aabbaa and abbbba, so T(6,2) = 2.
(In this case, the structure abaaba is excluded because it is not primitive.)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..930

Columns 2-6 are A056481, A056482, A056483, A056484, A056485.
Partial row sums include A056476, A056477, A056478, A056479, A056480.
Row sums are A284841.
Cf. A284823.

T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*StirlingS2[Ceiling[#/2], k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, Floor[(n+1)/2]}] // Flatten (* Jean-François Alcover, Jun 12 2017, from 2nd formula *)

b(n,k) = sumdiv(n,d, moebius(n/d) * k^(ceil(d/2)));
a(n,k) = sum(j=0,k, b(n,k-j)*binomial(k,j)*(-1)^j)/k!;
for(n=1, 20, for(k=1, ceil(n/2), print1( a(n,k),", ");); print(););

T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284823(n, k-j)) / k!.

T(n, k) = Sum_{d | n} mu(n/d) * stirling2(ceiling(d/2), k).

A056476 Number of primitive (aperiodic) palindromic structures of length n using a maximum of two different symbols.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 2, 7, 6, 14, 12, 31, 27, 63, 56, 123, 120, 255, 238, 511, 495, 1015, 992, 2047, 2010, 4092, 4032, 8176, 8127, 16383, 16242, 32767, 32640, 65503, 65280, 131061, 130788, 262143, 261632, 524223, 523770, 1048575, 1047494, 2097151, 2096127, 4194162

Offset: 0

Marks R. Nester

nonn

Permuting the symbols will not change the structure.

a(n) = A056481(n) for n > 1. - Jonathan Frech, May 21 2021

			Example from _Jonathan Frech_, May 21 2021: (Start)
The a(9)=14 lexicographically earliest equivalence class members in the alphabet {0,1} are:
  000010000
  000101000
  000111000
  001000100
  001010100
  001101100
  001111100
  010000010
  010101010
  010111010
  011000110
  011010110
  011101110
  011111110
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Michael De Vlieger, Table of n, a(n) for n = 0..5000

Cf. A056458, A056481, A284826.

Table[DivisorSum[n, MoebiusMu[#]*2^Floor[(n/# - 1)/2] &], {n, 46}] (* Michael De Vlieger, May 21 2021 *)

a(n) = if(n==0, 1, sumdiv(n, d, moebius(d)*2^((n/d-1)\2))) \\ Andrew Howroyd, May 21 2021

from sympy import mobius, divisors
def A056476(n): return sum(mobius(n//d)<<(d-1>>1) for d in divisors(n, generator=True)) if n else 1 # Chai Wah Wu, Feb 18 2024

a(n) = Sum_{d|n} mu(d)*A016116(n/d-1) for n > 0.
a(n) = Sum_{k=1..2} A284826(n, k) for n > 0. - Andrew Howroyd, May 21 2021
a(n) = A056458(n)/2 for n>=1. - Alois P. Heinz, Feb 18 2025

Definition clarified by Jonathan Frech, May 21 2021

a(0)=1 prepended and a(32)-a(45) from Andrew Howroyd, May 21 2021

A056518 Number of primitive (period n) periodic palindromic structures using exactly two different symbols.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 7, 10, 14, 21, 31, 42, 63, 91, 123, 184, 255, 371, 511, 750, 1015, 1519, 2047, 3030, 4092, 6111, 8176, 12222, 16383, 24486, 32767, 49024, 65503, 98175, 131061, 196308, 262143, 392959, 524223, 785910, 1048575, 1572256, 2097151, 3144702, 4194162

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 2 of A285037.

Cf. A056481.

A056513(n)-A000007(n-1).

a(17)-a(45) from Andrew Howroyd, Apr 08 2017

cp's OEIS Frontend

A284826 Irregular triangle T(n,k) for 1 <= k <= (n+1)/2: T(n,k) = number of primitive (aperiodic) palindromic structures of length n using exactly k different symbols.

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A056476 Number of primitive (aperiodic) palindromic structures of length n using a maximum of two different symbols.

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A056518 Number of primitive (period n) periodic palindromic structures using exactly two different symbols.

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