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Search: id:A162484
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| A162484 |
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a(1) = 2, a(2) = 8; a(n) = 2 a(n - 1) + a(n - 2) - 4 Mod[n, 2] |
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+0
1
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| 2, 8, 14, 36, 82, 200, 478, 1156, 2786, 6728, 16238, 39204, 94642, 228488, 551614, 1331716, 3215042, 7761800, 18738638, 45239076, 109216786, 263672648, 636562078, 1536796804, 3710155682, 8957108168
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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a(n) is the number of perfect matchings of an edge-labeled 2 x n toroidal grid graph, or equivalently the number of domino tilings of a 2 x n toroidal grid.
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REFERENCES
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s-m. belcastro, Tilings of 2 x n Grids on Surfaces, preprint.
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FORMULA
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for n > 2, 1/2 ((1 + Sqrt[2])^n (2 - (-2 + Sqrt[2]) (-1 + Sqrt[2])^(2 Floor[n/2])) + (1 - Sqrt[2])^n (2 + (1 + Sqrt[2])^(2 Floor[n/2]) (2 + Sqrt[2]))) (from Mathematica's solution to the recurrence)
Pell(n) + Pell(n-2) + 2 Mod[n-1,2]
a(n)= 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) = 2*A100828(n-1). G.f.: -2*x*(-1-2*x+3*x^2+2*x^3)/((x-1)*(1+x)*(x^2+2*x-1)) [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 26 2009]
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EXAMPLE
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a(3) = 2 a(2) + a(1) - 4 Mod[3,2] = 2*8 + 2 - 4 = 14
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CROSSREFS
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A000129
Sequence in context: A031160 A046959 A086177 this_sequence A039660 A119752 A111001
Adjacent sequences: A162481 A162482 A162483 this_sequence A162485 A162486 A162487
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KEYWORD
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easy,nonn
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AUTHOR
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Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009
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