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A208536 Number of 5-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row 5 of A208535.

Also, row 5 (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 (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

PROG

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

CROSSREFS

Cf. A208535, A208537.

Sequence in context: A059238 A254832 A026695 * A253947 A260344 A262354

Adjacent sequences:  A208533 A208534 A208535 * A208537 A208538 A208539

KEYWORD

nonn,easy

AUTHOR

R. H. Hardin, Feb 27 2012

STATUS

approved

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Last modified February 17 18:46 EST 2019. Contains 320222 sequences. (Running on oeis4.)