cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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.

Original entry on oeis.org

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

Views

Author

Andrew Howroyd, Apr 03 2017

Keywords

Comments

Permuting the symbols will not change the structure.

Examples

			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.)

References

  • 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]

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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(););

Formula

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).

A056458 Number of primitive (aperiodic) palindromes using a maximum of two different symbols.

Original entry on oeis.org

2, 0, 2, 2, 6, 4, 14, 12, 28, 24, 62, 54, 126, 112, 246, 240, 510, 476, 1022, 990, 2030, 1984, 4094, 4020, 8184, 8064, 16352, 16254, 32766, 32484, 65534, 65280, 131006, 130560, 262122, 261576, 524286, 523264, 1048446, 1047540, 2097150, 2094988, 4194302, 4192254
Offset: 1

Views

Author

Marks R. Nester

Keywords

Links

Crossrefs

Cf. A016116, A056476.
Column 2 of A284823.

Programs

  • Maple
    f:= proc(n) local d; add(numtheory:-mobius(d)*2^floor((1+n/d)/2), d = numtheory:-divisors(n)) end proc:
map(f, [$1..50]); # Robert Israel, Feb 18 2025
  • PARI
    a(n) = sumdiv(n, d, moebius(d)*2^((1 + n/d)\2)); \\ Michel Marcus, Apr 24 2020
  • Python
    from sympy import mobius, divisors
def A056458(n): return sum(mobius(n//d)<<(1+d>>1) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 18 2024

Formula

a(n) = Sum_{d|n} mu(d)*A016116(1 + n/d).
a(n) = 2 * A056476(n). - Alois P. Heinz, Feb 18 2025

Extensions

More terms from Michel Marcus, Apr 24 2020

A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

Original entry on oeis.org

1, 1, 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: 0

Views

Author

Marks R. Nester

Keywords

Comments

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
Number of Lyndon compositions (aperiodic necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

Examples

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic Lyndon compositions begins:
  (1)  (2)  (3)  (4)    (5)    (6)      (7)
                 (112)  (113)  (114)    (115)
                        (122)  (1122)   (133)
                               (11112)  (223)
                                        (11113)
                                        (11212)
                                        (11122)
(End)

References

  • 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]

Links

Crossrefs

Row sums of A179317.
Cf. A008965, A025065, A056476, A059966, A242414, A285037, A317085, A317086, A317087, A318731.

Programs

  • Mathematica
    (* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a56503[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])];
a[n_] := DivisorSum[n, MoebiusMu[#] a56503[n/#]&];
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)
  • PARI
    a(n) = {if(n < 1, n==0, sumdiv(n, d, moebius(d)*(2 + d%2)*(2^(n/d\2)))/(4 - n%2))} \\ Andrew Howroyd, Sep 26 2019
  • PARI
    seq(n) = Vec(1 + (1/2)*sum(k=1, n, moebius(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - moebius(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)) + O(x*x^n))) \\ Andrew Howroyd, Sep 27 2019

Formula

a(n) = Sum_{d|n} mu(d)*A056503(n/d) for n > 0.
a(n) = Sum_{k=1..2} A285037(n, k). - Andrew Howroyd, Apr 08 2017
G.f.: 1 + (1/2)*Sum_{k>=1} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - mu(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)). - Andrew Howroyd, Sep 27 2019

Extensions

a(17)-a(45) from Andrew Howroyd, Apr 08 2017
a(0)=1 prepended by Andrew Howroyd, Sep 27 2019

A056481 Number of primitive (aperiodic) palindromic structures using exactly two different symbols.

Original entry on oeis.org

0, 0, 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

Views

Author

Marks R. Nester

Keywords

Comments

Permuting the symbols will not change the structure. Identical to A056476 for n>1.

References

  • 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]

Crossrefs

Column 2 of A284826.
Cf. A056463.

Programs

  • Python
    from sympy import mobius, divisors
def A056481(n): return sum(mobius(n//d)<<(d-1>>1) for d in divisors(n, generator=True)) if n>1 else 0 # Chai Wah Wu, Feb 18 2024

Formula

a(n) = A056476(n) - A000007(n) - A000007(n-1).

Extensions

More terms (using A056476) from Joerg Arndt, May 22 2021

A056482 Number of primitive (aperiodic) palindromic structures using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 6, 6, 25, 24, 90, 89, 301, 295, 965, 960, 3025, 2999, 9330, 9305, 28495, 28411, 86526, 86430, 261624, 261324, 788945, 788669, 2375101, 2374110, 7141686
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Permuting the symbols will not change the structure.

References

  • 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]

Crossrefs

Column 3 of A284826.
Cf. A056464.

Formula

A056477(n)-A056476(n).
Showing 1-5 of 5 results.

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