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A208537
Number of 7-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color.
10
0, 0, 18, 312, 2340, 11160, 39990, 117648, 299592, 683280, 1428570, 2783880, 5118828, 8964072, 15059070, 24408480, 38347920, 58619808, 87460002, 127695960, 182857140, 257298360, 356336838, 486403632, 655210200, 871930800, 1147401450
OFFSET
1,3
COMMENTS
This sequence would be better defined as a(n) = (n^7-n)/7 with offset 0, which is an integer by Fermat's little theorem. - N. J. A. Sloane, Nov 13 2023
Row 7 of A208535.
Also, row 7 (with different offset) of A074650. - Eric M. Schmidt, Dec 08 2017
REFERENCES
J. Jeffries, Differentiating by prime numbers, Notices Amer. Math. Soc., 70:11 (2023), 1772-1779.
LINKS
FORMULA
Empirical: a(n) = (1/7)*n^7 - 1*n^6 + 3*n^5 - 5*n^4 + 5*n^3 - 3*n^2 + (6/7)*n.
Empirical formula confirmed by Petros Hadjicostas, Nov 05 2017 (see A208535).
a(n+2) = delta(-n) = -delta(n) for n >= 0, where delta is the p-derivation over the integers with respect to prime p = 7. - Danny Rorabaugh, Nov 10 2017
From Colin Barker, Nov 11 2017: (Start)
G.f.: 6*x^3*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)
a(n) = ((n-1)^7 - (n-1))/7. (inspired by Hassler's formula in A208536) - Eric M. Schmidt, Dec 08 2017
EXAMPLE
All solutions for n=3:
..1...1...1...1...1...1...1...1...1...1...1...1...1...1...1...1...1...1
..2...2...2...3...2...2...2...2...2...3...3...3...2...2...2...2...2...2
..1...1...1...1...1...1...3...3...3...2...2...1...3...1...3...1...3...1
..2...3...2...3...3...3...2...1...2...3...1...3...2...2...1...3...1...2
..3...2...1...2...1...2...1...3...3...2...3...1...3...3...3...1...2...1
..1...3...3...3...2...1...3...1...2...3...2...3...1...2...2...3...3...2
..3...2...2...2...3...3...2...3...3...2...3...2...3...3...3...2...2...3
MATHEMATICA
A208537[n_]:=((n-1)^7-(n-1))/7; Array[A208537, 50] (* Paolo Xausa, Nov 14 2023 *)
PROG
(PARI) Vec(6*x^3*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4) / (1 - x)^8 + O(x^40)) \\ Colin Barker, Nov 11 2017
CROSSREFS
Cf. A208535.
Sequence in context: A226298 A368537 A321511 * A292299 A158532 A214995
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Feb 27 2012
STATUS
approved