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A208537 Number of 7-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row 7 of A208535.

Also, row 7 (with different offset) of A074650. - Eric M. Schmidt, Dec 08 2017

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

Wikipedia, p-derivation.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

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

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: A179121 A226298 A321511 * A292299 A158532 A214995

Adjacent sequences:  A208534 A208535 A208536 * A208538 A208539 A208540

KEYWORD

nonn,easy

AUTHOR

R. H. Hardin, Feb 27 2012

STATUS

approved

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Last modified February 22 15:52 EST 2019. Contains 320399 sequences. (Running on oeis4.)