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 1-10 of 13 results. Next

A285012 Irregular triangle read by rows: T(n,k) is the number of periodic palindromic structures of length n using exactly k different symbols, 1 <= k <= n/2 + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 6, 5, 1, 1, 7, 6, 1, 1, 13, 19, 7, 1, 1, 15, 25, 10, 1, 1, 25, 64, 43, 10, 1, 1, 31, 90, 65, 15, 1, 1, 50, 208, 220, 85, 13, 1, 1, 63, 301, 350, 140, 21, 1, 1, 99, 656, 1059, 618, 154, 17, 1, 1, 127, 966, 1701, 1050, 266, 28, 1
Offset: 1

Views

Author

Andrew Howroyd, Apr 07 2017

Keywords

Comments

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
Equivalently, the number of necklaces, 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
1   1
1   1
1   3    1
1   3    1
1   6    5     1
1   7    6     1
1  13   19     7     1
1  15   25    10     1
1  25   64    43    10     1
1  31   90    65    15     1
1  50  208   220    85    13    1
1  63  301   350   140    21    1
1  99  656  1059   618   154   17   1
1 127  966  1701  1050   266   28   1
1 197 2035  4803  4065  1488  258  21  1
1 255 3025  7770  6951  2646  462  36  1
1 391 6250 21105 24915 12857 3222 410 26 1
1 511 9330 34105 42525 22827 5880 750 45 1
...
Example for n=6, k=2:
Periodic symmetry means results are either in the form abccba or abcdcb.
There are 3 binary words in the form abccba that start with 0 and contain a 1 which are 001100, 010010, 011110. Of these, 011110 is equivalent to 001100 after rotation.
There are 7 binary words in the form abcdcb that start with 0 and contain a 1 which are 000100, 001010, 001110, 010001, 010101, 011011, 011111. Of these, 011111 is equivalent to 000100, 010001 is equivalent to 001010 and 011011 is equivalent to 010010 from the first set.
There are therefore a total of 7 + 3 - 4 = 6 equivalence classes so T(6,2) = 6.

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 A056508, A056509, A056510, A056511, A056512.
Partial row sums include A056503, A056504, A056505, A056506, A056507.
Row sums are A285013.
Cf. A284855, A284826, A284823, A284877, A285037, A304972.

Programs

  • PARI
    \\ Ach is A304972, Prim is A285037.
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}
Prim(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) )))}
T(n,k=n\2+1) = {my(A=Prim(n,k)); Mat(vectorv(n, n, sumdiv(n, d, A[d, ])))}
{ my(A=T(20)); for(n=1, matsize(A)[1], print(A[n,1..n\2+1])) } \\ Andrew Howroyd, Oct 02 2019
  • PARI
    \\ column sequence using above code.
ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ Andrew Howroyd, Oct 02 2019

Formula

Column k is inverse Moebius transform of column k of A285037. - Andrew Howroyd, Oct 02 2019

A309784 T(n,k) is the number of non-equivalent distinguishing coloring partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 2, 1, 0, 1, 8, 10, 3, 1, 0, 1, 25, 32, 16, 3, 1, 0, 4, 62, 129, 84, 27, 4, 1, 0, 7, 176, 468, 433, 171, 37, 4, 1, 0, 18, 470, 1806, 2260, 1248, 338, 54, 5, 1, 0, 31, 1311, 6780, 11515, 8388, 3056, 590, 70, 5, 1, 0, 70, 3620, 25917, 58312, 56065, 26695, 6907, 1014, 96, 6, 1
Offset: 1

Views

Author

Mohammad Hadi Shekarriz, Aug 17 2019

Keywords

Comments

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. A distinguishing coloring partition of a graph G is a partition of the vertices of G such that it induces a distinguishing coloring for G. We say two distinguishing coloring partitions P1 and P2 of G are equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. Given a graph G, we use the notation psi_k(G) to denote the number of non-equivalent distinguishing coloring partitions of G with exactly k parts. For n>=3, this sequence gives T(n,k) = psi_k(C_n), i.e., the number of non-equivalent distinguishing coloring partitions of the cycle C_n on n vertices with exactly k parts.
T(n,k) is the number of primitive (period n) n-bead bracelet structures which are not periodic palindromes using exactly k different colored beads. - Andrew Howroyd, Sep 20 2019

Examples

			The triangle begins:
  0;
  0,  0;
  0,  0,   1;
  0,  0,   1,    1;
  0,  0,   4,    2,    1;
  0,  1,   8,   10,    3,    1;
  0,  1,  25,   32,   16,    3,   1;
  0,  4,  62,  129,   84,   27,   4,  1;
  0,  7, 176,  468,  433,  171,  37,  4, 1;
  0, 18, 470, 1806, 2260, 1248, 338, 54, 5, 1;
  ...
For n=6, we can partition the vertices of C_6 into exactly 3 parts in 8 ways such that all these partitions induce distinguishing colorings for C_6 and that all the 8 partitions are non-equivalent. The partitions are as follows:
    { { 1 }, { 2 }, { 3, 4, 5, 6 } }
    { { 1 }, { 2, 3 }, { 4, 5, 6 } }
    { { 1 }, { 2, 3, 4, 6 }, { 5 } }
    { { 1 }, { 2, 3, 5 }, { 4, 6 } }
    { { 1 }, { 2, 3, 6 }, { 4, 5 } }
    { { 1 }, { 2, 4, 5 }, { 3, 6 } }
    { { 1, 2 }, { 3, 4 }, { 5, 6 } }
    { { 1, 2 }, { 3, 5 }, { 4, 6 } }
For n=6, the above 8 partitions can be written as the following 3 colored bracelet structures: ABCCCC, ABBCCC, ABBBCB, ABBCBC, ABBCCB, ABCBBC, AABBCC, AABCBC. - _Andrew Howroyd_, Sep 22 2019

Links

Crossrefs

Column k=2 appears to be A011948.
Columns k=3..4 are A328038, A328039.
Row sums are A328035.
Cf. A107424, A152175, A152176, A276543, A276544, A285012, A285037, A309748, A309785.

Programs

  • PARI
    \\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(A=Ach(n), M=R(n), S=matrix(n, n, n, k, stirling(n, k, 2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(M[n/d,] + A[n/d,])/2 - 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(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Oct 02 2019

Formula

T(n,k) = A276543(n,k) - A285037(n,k). - Andrew Howroyd, Sep 20 2019

Extensions

T(10,6) corrected by Mohammad Hadi Shekarriz, Sep 28 2019
a(56)-a(78) from Andrew Howroyd, Sep 28 2019

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

A056514 Number of primitive (period n) periodic palindromic structures using a maximum of three different symbols.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 13, 28, 39, 84, 121, 244, 364, 741, 1088, 2200, 3280, 6591, 9841, 19720, 29510, 59169, 88573, 177240, 265716, 531804, 797121, 1594684, 2391484, 4783968, 7174453, 14350000, 21523238, 43050000, 64570064, 129143196, 193710244, 387430329, 581130368, 1162271280
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.

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

Cf. A056477, A056504, A285037.

Formula

a(n) = Sum_{d|n} mu(d)*A056504(n/d) for n > 0.
Moebius transform of A056504.
a(n) = Sum_{k=1..3} A285037(n, k) for n > 0. - Andrew Howroyd, Apr 08 2017

Extensions

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006
a(17)-a(35) from Andrew Howroyd, Apr 08 2017
a(0)=1 prepended and terms a(36) and beyond from Andrew Howroyd, Oct 01 2019

A056515 Number of primitive (period n) periodic palindromic structures using a maximum of four different symbols.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 10, 14, 35, 49, 127, 186, 463, 714, 1799, 2789, 6996, 11050, 27685, 43946, 109925, 175259, 438495, 700074, 1750445, 2798246, 6996927, 11188856, 27973533, 44747434, 111873782, 178973354, 447438656, 715860463, 1789673215, 2863377048, 7158463662
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.

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

Cf. A056478, A056505, A285037.

Formula

a(n) = Sum_{d|n} mu(d)*A056505(n/d) for n > 0.
Moebius transform of A056505.
a(n) = Sum_{k=1..4} A285037(n, k) for n > 0. - Andrew Howroyd, Oct 01 2019

Extensions

a(0)=1 prepended and terms a(17) and beyond from Andrew Howroyd, Oct 01 2019

A056516 Number of primitive (period n) periodic palindromic structures using a maximum of five different symbols.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 10, 14, 36, 50, 137, 201, 548, 854, 2417, 3839, 11060, 18001, 52599, 86471, 254982, 421989, 1252695, 2079474, 6196990, 10306747, 30795387, 51263890, 153409228, 255514354, 765389950, 1275163904, 3821990040, 6368612099, 19095299549, 31821472593
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.

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

Cf. A056479, A056506, A285037.

Formula

a(n) = Sum_{d|n} mu(d)*A056506(n/d) for n > 0.
Moebius transform of A056506. - T. D. Noe, Oct 25 2006
a(n) = Sum_{k=1..5} A285037(n, k) for n > 0. - Andrew Howroyd, Oct 01 2019

Extensions

Corrected by T. D. Noe, Oct 25 2006
a(0)=1 prepended and terms a(17) and beyond from Andrew Howroyd, Oct 01 2019

A056519 Number of primitive (period n) periodic palindromic structures using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 18, 25, 63, 90, 202, 301, 650, 965, 2016, 3025, 6220, 9330, 18970, 28495, 57650, 86526, 174210, 261624, 525693, 788945, 1582462, 2375101, 4759482, 7141686, 14300976, 21457735, 42951825, 64439003, 128946888, 193448101, 387037370, 580606145, 1161485370
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 A285037.
Cf. A056482, A056509, A056513, A056514.

Formula

a(n) = A056514(n) - A056513(n).
Moebius transform of A056509. - Andrew Howroyd, Oct 01 2019

Extensions

a(17)-a(35) from Andrew Howroyd, Apr 08 2017
Terms a(36) and beyond from Andrew Howroyd, Oct 01 2019

A056521 Number of primitive (period n) periodic palindromic structures using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 7, 10, 43, 65, 219, 350, 1058, 1701, 4796, 7770, 21094, 34105, 90205, 145749, 379326, 611501, 1573205, 2532530, 6465123, 10391735, 26378849, 42355950, 107089814, 171798901, 433088656, 694337225, 1746623215, 2798806984, 7029320466, 11259666950
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 4 of A285037.
Cf. A056483, A056510, A056514, A056515.

Formula

a(n) = A056515(n) - A056514(n).
Moebius transform of A056510. - T. D. Noe, Oct 25 2006

Extensions

Corrected by T. D. Noe, Oct 25 2006
a(17)-a(30) from Andrew Howroyd, Apr 08 2017
Terms a(31) and beyond from Andrew Howroyd, Oct 01 2019

A285042 Number of primitive (period n) periodic palindromic structures of length n using an infinite alphabet.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 10, 14, 36, 50, 138, 202, 562, 876, 2589, 4134, 12828, 21146, 69115, 115974, 398619, 678554, 2450202, 4213596, 15939338, 27644432, 109304036, 190899270, 787013630, 1382958544, 5931819804, 10480142146, 46673246440, 82864869600, 382473261356
Offset: 0

Views

Author

Andrew Howroyd, Apr 08 2017

Keywords

Comments

See A285037 for additional information. Permuting the symbols will not change the structure.

Links

Crossrefs

Row sums of A285037.
Cf. A285013.

Programs

  • PARI
    \\ Requires T from A285037.
seq(n)={my(A=T(n)); concat([1], vector(n, i, vecsum(A[i, ])))} \\ Andrew Howroyd, Oct 02 2019

Formula

a(n) = Sum_{d | n} mu(n/d) * A285013(d) for n > 0.

Extensions

a(0)=1 prepended and terms a(28) and beyond from Andrew Howroyd, Oct 02 2019

A056517 Number of primitive (period n) periodic palindromic structures using a maximum of six different symbols.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 10, 14, 36, 50, 138, 202, 561, 875, 2571, 4105, 12548, 20647, 65456, 109298, 356466, 601476, 2005341, 3403126, 11522216, 19628059, 67182654, 114700263, 395494722, 676207627, 2343260789, 4010090462, 13942474064, 23874361996, 83192218842, 142508723632
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.

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

Cf. A056480, A056507, A285037.

Formula

a(n) = Sum_{d|n} mu(d)*A056507(n/d) for n > 0.
a(n) = Sum_{k=1..6} A285037(n, k) for n > 0. - Andrew Howroyd, Oct 01 2019

Extensions

Corrected by T. D. Noe, Oct 25 2006
a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Oct 01 2019
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