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A332647
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a(n) = 2*a(n-1) + a(n-3) with a(0) = 3, a(1) = 2, a(2) = 4.
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2
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3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008, 13251, 29226, 64460, 142171, 313568, 691596, 1525363, 3364294, 7420184, 16365731, 36095756, 79611696, 175589123, 387274002, 854159700, 1883908523, 4155091048, 9164341796, 20212592115, 44580275278, 98324892352
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OFFSET
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0,1
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COMMENTS
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a(n) is the number of ways to tile a bracelet of length n with black trominos, and black or white squares.
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Washington & Lee University (2021).
Helmut Prodinger, On third-order Pell polynomials, arXiv:2011.04388 [math.NT], 2020.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).
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FORMULA
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a(n) = 2*a(n-1) + a(n-3).
a(n) = w1^n + w2^n + w3^n where w1,w2,w3 are the three roots of x^3-2x^2-1=0.
For n>2, a(n) = round(w1^n) for w1 the single real root of x^3-2x^2-1=0.
G.f.: (3 - 4*x) / (1 - 2*x - x^3). - Colin Barker, Feb 18 2020
a(n) = A008998(n) + 2*A008998(n-3) = 3*A008998(n) - 4*A008998(n-1).
a(n) = (5*b(n) - b(n-1) - b(n-2))/2 where b(n) = A052980(n). - Greg Dresden, Mar 10 2020
a(n) = A080204(n) + 1. - Greg Dresden, May 27 2020
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MATHEMATICA
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LinearRecurrence[{2, 0, 1}, {3, 2, 4}, 50]
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PROG
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(MAGMA) a:=[3, 2, 4]; [n le 3 select a[n] else 2*Self(n-1)+Self(n-3):n in [1..33]]; // Marius A. Burtea, Feb 18 2020
(PARI) Vec((3 - 4*x) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Feb 18 2020
(PARI) polsym(x^3-2*x^2-1, 44) \\ Joerg Arndt, May 28 2020
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CROSSREFS
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Cf. A008998, A052980. Equals one more than A080204.
Sequence in context: A296099 A349853 A319103 * A290333 A137824 A019321
Adjacent sequences: A332644 A332645 A332646 * A332648 A332649 A332650
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KEYWORD
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easy,nonn
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AUTHOR
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Greg Dresden, Feb 18 2020
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STATUS
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approved
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