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A162484 a(1) = 2, a(2) = 8; a(n) = 2 a(n - 1) + a(n - 2) - 4*(n mod 2). 1
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, 21624372014, 52205852196, 126036076402, 304278005000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

S-M. Belcastro, Tilings of 2 x n Grids on Surfaces, preprint.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

FORMULA

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*((n-1) mod 2).

From R. J. Mathar, Jul 26 2009: (Start)

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)).

(End)

a(n) = 1 + (-1)^n + (1-sqrt(2))^n + (1+sqrt(2))^n. - Colin Barker, Dec 16 2017

EXAMPLE

a(3) = 2 a(2) + a(1) - 4*(3 mod 2) = 2*8 + 2 - 4 = 14.

MATHEMATICA

Fold[Append[#1, 2 #1[[#2 - 1]] + #1[[#2 - 2]] - 4 Mod[#2, 2]] &, {2, 8}, Range[3, 30]] (* or *)

Rest@ CoefficientList[Series[-2 x (-1 - 2 x + 3 x^2 + 2 x^3)/((x - 1) (1 + x) (x^2 + 2 x - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Dec 16 2017 *)

LinearRecurrence[{2, 2, -2, -1}, {2, 8, 14, 36}, 30] (* Harvey P. Dale, Aug 24 2018 *)

CROSSREFS

Cf. A000129.

Sequence in context: A046959 A086177 A299337 * A259119 A039660 A119752

Adjacent sequences:  A162481 A162482 A162483 * A162485 A162486 A162487

KEYWORD

easy,nonn

AUTHOR

Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009

STATUS

approved

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Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)