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A162483 a(n) is the number of perfect matchings of an edge-labeled 2 X (2n+1) Mobius grid graph. 1
3, 6, 13, 31, 78, 201, 523, 1366, 3573, 9351, 24478, 64081, 167763, 439206, 1149853, 3010351, 7881198, 20633241, 54018523, 141422326, 370248453, 969323031, 2537720638, 6643838881, 17393796003, 45537549126, 119218851373, 312119004991, 817138163598 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is a specialization for m=2 of a general formula for the number of perfect matchings of an edge-labeled m X (2n+1) Mobius grid graph.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

FORMULA

a(n) = Real((1-I) * (L(2*n+1) - F(2*n+1))/2 + F(2*n+2) + 2*I)).

From R. J. Mathar, Aug 08 2009: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).

G.f.: (3-6*x+x^2)/((1-x)*(x^2-3*x+1)). (End)

a(n+1)-a(n) = A005248(n+1). - R. J. Mathar, Dec 18 2010

a(n) = A000032(2n-1)+2. - Clark Kimberling, Oct 26 2012

a(n) = 2^(-1-n)*(2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n). - Colin Barker, Nov 03 2016

a(n) = 2 + L(2*n+1), A256233(n) = -a(-n-1) for all n in Z. - Michael Somos, Nov 03 2016

EXAMPLE

G.f. = 3 + 6*x + 13*x^2 + 31*x^3 + 78*x^4 + 201*x^5 + 523*x^6 + 1366*x^7 + ...

a(0) = 3 because this is the number of perfect matchings of a 2 X 1 Mobius grid graph (one for each of the three multiple edges).

MATHEMATICA

Table[Re[(1 - I) (2*I + Fibonacci[2 + 2*n] + 1/2 (-Fibonacci[1 + 2*n] + LucasL[1 + 2*n]))], {n, 0, 30}]

Table[LucasL[2*n + 1] + 2, {n, 0, 30}] (* Clark Kimberling, Oct 26 2012 *)

LinearRecurrence[{4, -4, 1}, {3, 6, 13}, 30] (* or *) CoefficientList[Series[(-3 + 6 x - x^2)/(-1 + 4 x - 4 x^2 + x^3), {x, 0, 30}], x] (* Stefano Spezia, Sep 23 2018 *)

PROG

(PARI) {a(n) = 2 + fibonacci(2*n) + fibonacci(2*n+2)}; /* Michael Somos, Nov 03 2016 */

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((3-6*x+x^2)/((1-x)*(x^2-3*x+1)))); // G. C. Greubel, Sep 22 2018

CROSSREFS

Cf. A020878, A256233.

Sequence in context: A126296 A293911 A018014 * A187780 A273974 A179928

Adjacent sequences:  A162480 A162481 A162482 * A162484 A162485 A162486

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified May 24 15:25 EDT 2019. Contains 323532 sequences. (Running on oeis4.)