A208536 Number of 5-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color.

0, 0, 6, 48, 204, 624, 1554, 3360, 6552, 11808, 19998, 32208, 49764, 74256, 107562, 151872, 209712, 283968, 377910, 495216, 639996, 816816, 1030722, 1287264, 1592520, 1953120, 2376270, 2869776, 3442068, 4102224, 4859994, 5725824, 6710880

Offset

1,3

Comments

This sequence would be better defined as a(n) = (n^5-n)/5 with offset 0, which is an integer by Fermat's little theorem. - N. J. A. Sloane, Nov 13 2023

Links

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

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

Wikipedia, p-derivation.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

Empirical: a(n) = (1/5)*n^5 - 1*n^4 + 2*n^3 - 2*n^2 + (4/5)*n.

Equivalently: a(n) = ((n-1)^5 - (n-1))/5. - M. F. Hasler, Mar 05 2016

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 = 5. - Danny Rorabaugh, Nov 10 2017

From Colin Barker, Nov 11 2017: (Start)

G.f.: 6*x^3*(1 + x)^2 / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

(End)

Example

All solutions for n=3:
..1....1....1....1....1....1
..3....3....2....2....2....2
..1....2....1....3....3....1
..3....3....3....2....1....2
..2....2....2....3....3....3

Mathematica

A208536[n_]:=((n-1)^5-(n-1))/5; Array[A208536, 50] (* Paolo Xausa, Nov 14 2023 *)

Prog

(PARI) Vec(6*x^3*(1 + x)^2 / (1 - x)^6 + O(x^40)) \\ Colin Barker, Nov 11 2017

Crossrefs

Row 5 of A208535.

Also, row 5 (with different offset) of A074650. - Eric M. Schmidt, Dec 08 2017

Cf. A208537.

Sequence in context: A371067 A254832 A026695 * A253947 A260344 A353247

Adjacent sequences: A208533 A208534 A208535 * A208537 A208538 A208539

Keyword

nonn,easy

Author

R. H. Hardin, Feb 27 2012

Status

approved