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A208536
Number of 5-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color.
10
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
Jack Jeffries, Differentiating by prime numbers, Notices Amer. Math. Soc., 70:11 (2023), 1772-1779.
Wikipedia, p-derivation.
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
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Feb 27 2012
STATUS
approved