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
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;
...
-
T(n,k) = {sumdiv(n, d, moebius(n/d)*k!*stirling(ceil(d/2), k, 2))}
A056498
Number of primitive (period n) periodic palindromes using exactly two different symbols.
Original entry on oeis.org
0, 1, 2, 3, 6, 7, 14, 18, 28, 39, 62, 81, 126, 175, 246, 360, 510, 728, 1022, 1485, 2030, 3007, 4094, 6030, 8184, 12159, 16352, 24381, 32766, 48849, 65534, 97920, 131006, 196095, 262122, 392364, 524286, 785407, 1048446, 1571310, 2097150, 3143497, 4194302, 6288381
Offset: 1
- 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]
-
seq(n)={Vec(sum(k=1, n\2, moebius(k)*x^(2*k)*(1 + x^k)/((1 - x^k)*(1 - 2*x^(2*k))) + O(x*x^n)), -n)} \\ Andrew Howroyd, Sep 29 2019
A327879
Number of primitive (period n) periodic palindromes with integer entries that cover an initial interval of positive integers.
Original entry on oeis.org
1, 1, 1, 2, 6, 12, 40, 74, 300, 538, 2598, 4682, 25938, 47292, 296488, 545820, 3816240, 7087260, 54666830, 102247562, 862437450, 1622632496, 14857095400, 28091567594, 277474931700, 526858348368, 5584100612118, 10641342969902, 120462266677578
Offset: 0
The a(4) = 6 primitive periodic palindromes are:
1122, 1112, 1222,
1213, 1232, 1323.
-
a(n)={if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=0, n, k!*(stirling((d+1)\2, k, 2)+stirling(d\2+1, k, 2))))/2)}
A056499
Number of primitive (period n) periodic palindromes using exactly three different symbols.
Original entry on oeis.org
0, 0, 0, 3, 6, 21, 36, 90, 150, 339, 540, 1149, 1806, 3765, 5790, 11880, 18150, 36894, 55980, 113145, 170970, 344541, 519156, 1043190, 1569744, 3149979, 4733670, 9488409, 14250606, 28544205, 42850116, 85786560, 128746410, 257672355, 386634018, 773623116, 1160688606
Offset: 1
- 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]
A056500
Number of primitive (period n) periodic palindromes using exactly four different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 12, 24, 132, 240, 900, 1560, 4968, 8400, 24588, 40824, 113520, 186480, 502248, 818520, 2157360, 3497976, 9085452, 14676024, 37723260, 60780720, 155082900, 249401640, 632947728, 1016542800, 2569816476, 4123173624, 10393520640
Offset: 1
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
- 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]
A056501
Number of primitive (period n) periodic palindromes using exactly five different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 60, 120, 960, 1800, 9300, 16800, 71400, 126000, 480000, 834120, 2968440, 5103000, 17354340, 29607600, 97566000, 165528000, 533264700, 901020120, 2854995360, 4809004080, 15050445900, 25292030400, 78417321240, 131542866000, 404936052000
Offset: 1
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
- 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]
A056502
Number of primitive (period n) periodic palindromes using exactly six different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 360, 720, 7920, 15120, 103320, 191520, 1048320, 1905120, 9170280, 16435440, 72832680, 129230640, 541129320, 953029440, 3832179120, 6711344640, 26192751480, 45674188560, 174286569240, 302899156560, 1136022947280, 1969147121760
Offset: 1
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
- 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]
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