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 15 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

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).

A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62
Offset: 1

Views

Author

Andrew Howroyd, Apr 03 2017

Keywords

Examples

			Table starts:
1  2   3    4    5    6     7     8     9    10 ...
0  0   0    0    0    0     0     0     0     0 ...
0  2   6   12   20   30    42    56    72    90 ...
0  2   6   12   20   30    42    56    72    90 ...
0  6  24   60  120  210   336   504   720   990 ...
0  4  18   48  100  180   294   448   648   900 ...
0 14  78  252  620 1290  2394  4088  6552  9990 ...
0 12  72  240  600 1260  2352  4032  6480  9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216  960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).

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 A056458, A056459, A056460, A056461, A056462.
Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
Cf. A284826, A284841.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)
  • PARI
    a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)

Formula

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

A284877 Irregular triangle T(n,k) for 1 <= k <= n/2 + 1: T(n,k) = number of double palindrome structures of length n using exactly k different symbols.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 2, 1, 15, 5, 1, 25, 21, 3, 1, 49, 42, 7, 1, 79, 122, 44, 4, 1, 129, 225, 90, 9, 1, 211, 570, 375, 80, 5, 1, 341, 990, 715, 165, 11, 1, 517, 2321, 2487, 930, 132, 6, 1, 819, 3913, 4550, 1820, 273, 13, 1, 1275, 8827, 14350, 8330, 2009, 203, 7
Offset: 1

Views

Author

Andrew Howroyd, Apr 04 2017

Keywords

Comments

A double palindrome is the concatenation of two palindromes. Permuting the symbols will not change the structure. For the purposed of this sequence, valid palindromes include both the empty word and a singleton symbol.

Examples

			Triangle starts:
1
1     1
1     3
1     7      2
1    15      5
1    25     21       3
1    49     42       7
1    79    122      44       4
1   129    225      90       9
1   211    570     375      80       5
1   341    990     715     165      11
1   517   2321    2487     930     132      6
1   819   3913    4550    1820     273     13
1  1275   8827   14350    8330    2009    203      7
1  1863  14480   25515   15750    3990    420     15
1  2959  31802   75724   64004   23296   3920    296     8
1  4335  51425  132090  118167   44982   7854    612    17
1  6703 110928  376779  445275  229257  57078   7074   414   9
1  9709 177270  647995  807975  433713 111720  14250   855  19
1 15067 377722 1798175 2892470 2023135 698670 126300 12000 560 10
....
The first few structures are:
n = 1: a => 1
n = 2: aa; ab => 1 + 1
n = 3: aaa; aab, aba, abb => 1 + 3
n = 4: aaaa; aaab, aaba, aabb, abaa, abab, abba, abbb; abac, abcb => 1 + 7 + 2

Links

Crossrefs

Columns k=2..4 are A328688, A328689, A328690.
Row sums are A165137.
Partial row sums include A180249, A328692, A328693.
Cf. A284873, A284826.

Programs

  • Mathematica
    r[d_, k_]:=If[OddQ[d], d*k^((d + 1)/2), (d/2)*(k + 1)*k^(d/2)]; a[n_, k_]:= r[n, k] - Sum[If[dIndranil Ghosh, Apr 07 2017 *)
  • PARI
    r(d,k)=if (d % 2 == 0, (d/2)*(stirling(d/2,k,2)+stirling(d/2+1,k,2)), d*stirling((d+1)/2, k,2));
a(n,k) = r(n,k) - sumdiv(n,d, if (d
  • Python
    from sympy import totient, divisors, binomial, factorial
def r(d, k): return (d//2)*(k + 1)*k**(d//2) if d%2 == 0 else d*k**((d + 1)//2)
def a(n, k): return r(n, k) - sum([totient(n//d)*a(d, k) for d in divisors(n) if dIndranil Ghosh, Apr 07 2017

Formula

T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284873(n, k-j)) / k!.
T(n, k) = r(n, k) - Sum_{d|n, d

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

Views

Author

Marks R. Nester

Keywords

Comments

Permuting the symbols will not change the structure.
a(n) = A056481(n) for n > 1. - Jonathan Frech, May 21 2021

Examples

			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)

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. A056458, A056481, A284826.

Programs

  • Mathematica
    Table[DivisorSum[n, MoebiusMu[#]*2^Floor[(n/# - 1)/2] &], {n, 46}] (* Michael De Vlieger, May 21 2021 *)
  • PARI
    a(n) = if(n==0, 1, sumdiv(n, d, moebius(d)*2^((n/d-1)\2))) \\ Andrew Howroyd, May 21 2021
  • Python
    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

Formula

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

Extensions

Definition clarified by Jonathan Frech, May 21 2021
a(0)=1 prepended and a(32)-a(45) from Andrew Howroyd, May 21 2021

A056477 Number of primitive (aperiodic) palindromic structures using a maximum of three different symbols.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 3, 13, 12, 39, 36, 121, 116, 364, 351, 1088, 1080, 3280, 3237, 9841, 9800, 29510, 29403, 88573, 88440, 265716, 265356, 797121, 796796, 2391484, 2390352, 7174453, 7173360, 21523238, 21520080, 64570064, 64566684, 193710244, 193700403, 581130368, 581120880
Offset: 0

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]

Links

Crossrefs

Cf. A007051, A124302, A056459, A284826.

Formula

a(n) = Sum_{d|n} mu(d)*A124302(ceiling(n/(2*d))) for n > 0.
a(n) = Sum_{k=1..3} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

Extensions

a(0)=1 prepended and terms a(32) and beyond from Andrew Howroyd, Oct 02 2019

A056478 Number of primitive (aperiodic) palindromic structures using a maximum of four different symbols.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 3, 14, 13, 49, 46, 186, 181, 714, 700, 2789, 2780, 11050, 10997, 43946, 43895, 175259, 175088, 700074, 699875, 2798246, 2797536, 11188856, 11188191, 44747434, 44744591, 178973354, 178970560, 715860463, 715849600, 2863377048, 2863365834
Offset: 0

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]

Links

Crossrefs

Cf. A007581, A124303, A056460, A284826.

Formula

a(n) = Sum_{d|n} mu(d)*A124303(ceiling(n/(2*d))) for n > 0.
a(n) = Sum_{k=1..4} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

Extensions

a(0)=1 prepended and terms a(32) and beyond from Andrew Howroyd, Oct 02 2019

A056479 Number of primitive (aperiodic) palindromic structures using a maximum of five different symbols.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 3, 14, 13, 50, 47, 201, 196, 854, 840, 3839, 3830, 18001, 17947, 86471, 86419, 421989, 421803, 2079474, 2079260, 10306747, 10305897, 51263890, 51263086, 255514354, 255510460, 1275163904, 1275160060, 6368612099, 6368594300, 31821472593, 31821454413
Offset: 0

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]

Links

Crossrefs

Cf. A056461, A056470, A284826.

Formula

a(n) = Sum_{d|n} mu(d)*A056470(n/d) for n > 0.
a(n) = Sum_{k=1..5} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

Extensions

a(0)=1 prepended and terms a(32) and beyond from Andrew Howroyd, Oct 02 2019

A056480 Number of primitive (aperiodic) palindromic structures using a maximum of six different symbols.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 3, 14, 13, 50, 47, 202, 197, 875, 861, 4105, 4096, 20647, 20593, 109298, 109246, 601476, 601289, 3403126, 3402911, 19628059, 19627188, 114700263, 114699438, 676207627, 676203467, 4010090462, 4010086352, 23874361996, 23874341552, 142508723632
Offset: 0

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]

Links

Crossrefs

Cf. A056462, A056471, A284826.

Formula

a(n) = Sum_{d|n} mu(d)*A056471(n/d) for n > 0.
a(n) = Sum_{k=1..6} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

Extensions

a(0)=1 prepended and terms a(32) and beyond from Andrew Howroyd, Oct 02 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
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