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 1-3 of 3 results.

A285037 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromic structures using exactly k different symbols, 1 <= k <= n/2 + 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 1, 0, 3, 1, 0, 4, 5, 1, 0, 7, 6, 1, 0, 10, 18, 7, 1, 0, 14, 25, 10, 1, 0, 21, 63, 43, 10, 1, 0, 31, 90, 65, 15, 1, 0, 42, 202, 219, 85, 13, 1, 0, 63, 301, 350, 140, 21, 1, 0, 91, 650, 1058, 618, 154, 17, 1, 0, 123, 965, 1701, 1050, 266, 28, 1
Offset: 1

Views

Author

Andrew Howroyd, Apr 08 2017

Keywords

Comments

Permuting the symbols will not change the structure.
Equivalently, the number of n-bead aperiodic necklaces (Lyndon words) with exactly k symbols, up to permutation of the symbols, which when turned over are unchanged. When comparing with the turned over necklace a rotation is allowed but a permutation of the symbols is not.

Examples

			Triangle starts:
1
0   1
0   1
0   2    1
0   3    1
0   4    5     1
0   7    6     1
0  10   18     7     1
0  14   25    10     1
0  21   63    43    10     1
0  31   90    65    15     1
0  42  202   219    85    13    1
0  63  301   350   140    21    1
0  91  650  1058   618   154   17   1
0 123  965  1701  1050   266   28   1
0 184 2016  4796  4064  1488  258  21  1
0 255 3025  7770  6951  2646  462  36  1
0 371 6220 21094 24914 12857 3222 410 26 1
0 511 9330 34105 42525 22827 5880 750 45 1
...
Example for n=6, k=2:
There are 6 inequivalent solutions to A285012(6,2) which are 001100, 010010, 000100, 001010, 001110, 010101. Of these, 010010 and 010101 have a period less than 6, so T(6,2) = 6-2 = 4.

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 1..6 are: A063524, A056518, A056519, A056521, A056522, A056523.
Partial row sums include A056513, A056514, A056515, A056516, A056517.
Row sums are A285042.
Cf. A284856, A284826, A284823, A285012, A304972.

Programs

  • PARI
    \\ Ach is A304972
Ach(n,k=n) = {my(M=matrix(n, k, n, k, n>=k)); for(n=3, n, for(k=2, k, M[n, k]=k*M[n-2, k] + M[n-2, k-1] + if(k>2, M[n-2, k-2]))); M}
T(n,k=n\2+1) = {my(A=Ach(n\2+1,k), S=matrix(n\2+1, k, n, k, stirling(n,k,2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}
{ my(A=T(20)); for(n=1, matsize(A)[1], print(A[n,1..n\2+1])) } \\ Andrew Howroyd, Oct 01 2019
  • PARI
    \\ column sequence using above code.
ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ Andrew Howroyd, Oct 01 2019

Formula

T(n, k) = Sum_{d | n} mu(n/d) * A285012(d, k).

A056509 Number of periodic palindromic structures of length n using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 19, 25, 64, 90, 208, 301, 656, 966, 2035, 3025, 6250, 9330, 19035, 28501, 57740, 86526, 174436, 261625, 525994, 788970, 1583119, 2375101, 4760516, 7141686, 14303011, 21457825, 42954850, 64439010, 128953341, 193448101, 387046700, 580606446, 1161504423
Offset: 1

Views

Author

Marks R. Nester

Keywords

Examples

			For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. 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]

Links

Crossrefs

Column 3 of A285012.
Cf. A000392, A056503, A056504, A056519.

Formula

a(n) = A056504(n) - A056503(n).
Inverse Moebius transform of A056519. - Andrew Howroyd, Oct 01 2019

Extensions

a(17)-a(35) from Andrew Howroyd, Apr 07 2017
Terms a(36) and beyond from Andrew Howroyd, Oct 01 2019

A328038 Number of primitive (period n) n-bead bracelet structures which are not periodic palindromes using exactly three different colored beads.

Original entry on oeis.org

0, 0, 1, 1, 4, 8, 25, 62, 176, 470, 1311, 3620, 10094, 28209, 79236, 223270, 631240, 1790213, 5090995, 14515788, 41484907, 118821599, 341008317, 980487770, 2823961866, 8146372122, 23534556225, 68083326558, 197209108054, 571910949743, 1660395053569, 4825540091342
Offset: 1

Views

Author

Andrew Howroyd, Oct 02 2019

Keywords

Comments

Permuting the colors of the beads will not change the structure.

Examples

			For n = 3, the 1 bracelet structure has the pattern: ABC.
For n = 4, the 1 bracelet structure has the pattern: AABC.
For n = 5, the 4 bracelet structures have the patterns: AAABC, AABAC, AABBC, ABABC. The pattern ABBAC is excluded because it is a periodic palindrome.
For n = 6, the 8 bracelet structures have the patterns: ABCCCC, ABBCCC, ABBBCB, ABBCBC, ABBCCB, ABCBBC, AABBCC, AABCBC.

Links

Crossrefs

Column 3 of A309784.
Cf. A056367, A056519, A328035, A328039, A328657.

Formula

a(n) = A056367(n) - A056519(n).
Showing 1-3 of 3 results.

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