A284823 -id:A284823 - 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).

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

Andrew Howroyd, Apr 07 2017

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.

			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.

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..2600

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.

\\ 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

\\ column sequence using above code.
ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ Andrew Howroyd, Oct 02 2019

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

Andrew Howroyd, Apr 08 2017

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.

			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.

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..2600

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.

\\ 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

\\ column sequence using above code.
ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ Andrew Howroyd, Oct 01 2019

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

A327873 Irregular triangle read by rows: T(n,k) is the number of length n primitive (aperiodic) palindromes using exactly k different symbols, 1 <= k <= ceiling(n/2).

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 6, 6, 0, 4, 6, 0, 14, 36, 24, 0, 12, 36, 24, 0, 28, 150, 240, 120, 0, 24, 144, 240, 120, 0, 62, 540, 1560, 1800, 720, 0, 54, 534, 1560, 1800, 720, 0, 126, 1806, 8400, 16800, 15120, 5040, 0, 112, 1770, 8376, 16800, 15120, 5040

Offset: 1

Andrew Howroyd, Sep 28 2019

			Triangle begins:
  1;
  0;
  0,   2;
  0,   2;
  0,   6,    6;
  0,   4,    6;
  0,  14,   36,   24;
  0,  12,   36,   24;
  0,  28,  150,  240,   120;
  0,  24,  144,  240,   120;
  0,  62,  540, 1560,  1800,   720;
  0,  54,  534, 1560,  1800,   720;
  0, 126, 1806, 8400, 16800, 15120, 5040;
  0, 112, 1770, 8376, 16800, 15120, 5040;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2550

Columns k=2..6 are A056463, A056464, A056465, A056466, A056467.
Row sums are A327874.
Cf. A284823, A327878.

T(n,k) = {sumdiv(n, d, moebius(n/d)*k!*stirling(ceil(d/2), k, 2))}

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

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

Marks R. Nester

nonn

Robert Israel, Table of n, a(n) for n = 1..6610
M. R. Nester, Mathematical investigations of some plant interaction designs, PhD Thesis, (1999), University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A016116, A056476.

Column 2 of A284823.

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

a(n) = sumdiv(n, d, moebius(d)*2^((1 + n/d)\2)); \\ Michel Marcus, Apr 24 2020

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

a(n) = Sum_{d|n} mu(d)*A016116(1 + n/d).

a(n) = 2 * A056476(n). - Alois P. Heinz, Feb 18 2025

More terms from Michel Marcus, Apr 24 2020

A056459 Number of primitive (aperiodic) palindromes using a maximum of three different symbols.

Original entry on oeis.org

3, 0, 6, 6, 24, 18, 78, 72, 234, 216, 726, 696, 2184, 2106, 6528, 6480, 19680, 19422, 59046, 58800, 177060, 176418, 531438, 530640, 1594296, 1592136, 4782726, 4780776, 14348904, 14342112, 43046718, 43040160, 129139428, 129120480, 387420384, 387400104, 1162261464, 1162202418, 3486782208

Offset: 1

Marks R. Nester

nonn

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]

Cf. A056449.

Column 3 of A284823.

a(n) = sumdiv(n, d, moebius(d)*3^((1 + n/d)\2));
for(n=1, 40, print1(a(n), ",")); \\ Petros Hadjicostas, Apr 24 2020

a(n) = Sum_{d | n} mu(d)*A056449(n/d).

More terms from Petros Hadjicostas, Apr 24 2020

A056460 Number of primitive (aperiodic) palindromes using a maximum of four different symbols.

Original entry on oeis.org

4, 0, 12, 12, 60, 48, 252, 240, 1008, 960, 4092, 4020, 16380, 16128, 65460, 65280, 262140, 261072, 1048572, 1047540, 4194036, 4190208, 16777212, 16772880, 67108800, 67092480, 268434432, 268419060, 1073741820, 1073675280, 4294967292, 4294901760, 17179865076, 17179607040, 68719476420

Offset: 1

Marks R. Nester

nonn

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 4 of A284823.

Cf. A056450.

a(n) = sumdiv(n, d, moebius(d)*(3*2^(n/d)-(-2)^(n/d))/2);
for(n=1, 40, print1(a(n), ", ")); \\ Petros Hadjicostas, Apr 24 2020

a(n) = Sum_{d|n} mu(d)*A056450(n/d).

More terms from Petros Hadjicostas, Apr 24 2020

A056461 Number of primitive (aperiodic) palindromes using a maximum of five different symbols.

Original entry on oeis.org

5, 0, 20, 20, 120, 100, 620, 600, 3100, 3000, 15620, 15480, 78120, 77500, 390480, 390000, 1953120, 1949900, 9765620, 9762480, 48827480, 48812500, 244140620, 244124400, 1220703000, 1220625000, 6103512500, 6103437480, 30517578120, 30517184400, 152587890620, 152587500000

Offset: 1

Marks R. Nester

nonn

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]

Cf. A056391, A056451.

Column 5 of A284823.

a(n) = sumdiv(n, d, moebius(d)*5^((1 + n/d)\2));
for(n=1, 40, print1(a(n), ", ")); \\ Petros Hadjicostas, Apr 24 2020

Sum_{d|n} mu(d)*A056451(n/d).

More terms from Petros Hadjicostas, Apr 24 2020

A056462 Number of primitive (aperiodic) palindromes using a maximum of six different symbols.

Original entry on oeis.org

6, 0, 30, 30, 210, 180, 1290, 1260, 7740, 7560, 46650, 46410, 279930, 278640, 1679370, 1678320, 10077690, 10069740, 60466170, 60458370, 362795730, 362750400, 2176782330, 2176734420, 13060693800, 13060414080, 78364156320, 78363884130, 470184984570, 470183297220

Offset: 1

Marks R. Nester

nonn

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]

Cf. A056391, A056452.

Column 6 of A284823.

a(n) = sumdiv(n, d, moebius(d)*6^((1 + n/d)\2));
for(n=1, 32, print1(a(n), ", ")); \\ Petros Hadjicostas, Apr 24 2020

a(n) = Sum_{d|n} mu(d)*A056452(n/d).

More terms from Petros Hadjicostas, Apr 24 2020

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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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.

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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.

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A327873 Irregular triangle read by rows: T(n,k) is the number of length n primitive (aperiodic) palindromes using exactly k different symbols, 1 <= k <= ceiling(n/2).

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

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A056459 Number of primitive (aperiodic) palindromes using a maximum of three different symbols.

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A056460 Number of primitive (aperiodic) palindromes using a maximum of four different symbols.

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