A208537 - OEIS

%I #27 Nov 14 2023 17:03:42

%S 0,0,18,312,2340,11160,39990,117648,299592,683280,1428570,2783880,

%T 5118828,8964072,15059070,24408480,38347920,58619808,87460002,

%U 127695960,182857140,257298360,356336838,486403632,655210200,871930800,1147401450

%N Number of 7-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color.

%C 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

%C Row 7 of A208535.

%C Also, row 7 (with different offset) of A074650. - _Eric M. Schmidt_, Dec 08 2017

%D J. Jeffries, Differentiating by prime numbers, Notices Amer. Math. Soc., 70:11 (2023), 1772-1779.

%H R. H. Hardin, <a href="/A208537/b208537.txt">Table of n, a(n) for n = 1..210</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/P-derivation">p-derivation</a>.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F 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.

%F Empirical formula confirmed by _Petros Hadjicostas_, Nov 05 2017 (see A208535).

%F 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

%F From _Colin Barker_, Nov 11 2017: (Start)

%F G.f.: 6*x^3*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4) / (1 - x)^8.

%F 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.

%F (End)

%F a(n) = ((n-1)^7 - (n-1))/7. (inspired by Hassler's formula in A208536) - _Eric M. Schmidt_, Dec 08 2017

%e All solutions for n=3:

%e ..1...1...1...1...1...1...1...1...1...1...1...1...1...1...1...1...1...1

%e ..2...2...2...3...2...2...2...2...2...3...3...3...2...2...2...2...2...2

%e ..1...1...1...1...1...1...3...3...3...2...2...1...3...1...3...1...3...1

%e ..2...3...2...3...3...3...2...1...2...3...1...3...2...2...1...3...1...2

%e ..3...2...1...2...1...2...1...3...3...2...3...1...3...3...3...1...2...1

%e ..1...3...3...3...2...1...3...1...2...3...2...3...1...2...2...3...3...2

%e ..3...2...2...2...3...3...2...3...3...2...3...2...3...3...3...2...2...3

%t A208537[n_]:=((n-1)^7-(n-1))/7;Array[A208537,50] (* _Paolo Xausa_, Nov 14 2023 *)

%o (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

%Y Cf. A208535.

%K nonn,easy

%O 1,3

%A _R. H. Hardin_, Feb 27 2012