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A208536
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Number of 5-bead necklaces of n colors not allowing reversal, with no adjacent beads having the same color.
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6
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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
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OFFSET
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1,3
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COMMENTS
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Row 5 of A208535.
Also, row 5 (with different offset) of A074650. - Eric M. Schmidt, Dec 08 2017
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LINKS
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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).
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FORMULA
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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)
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EXAMPLE
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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
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PROG
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(PARI) Vec(6*x^3*(1 + x)^2 / (1 - x)^6 + O(x^40)) \\ Colin Barker, Nov 11 2017
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CROSSREFS
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Cf. A208535, A208537.
Sequence in context: A059238 A254832 A026695 * A253947 A260344 A262354
Adjacent sequences: A208533 A208534 A208535 * A208537 A208538 A208539
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KEYWORD
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nonn,easy
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AUTHOR
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R. H. Hardin, Feb 27 2012
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STATUS
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approved
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