A162483 a(n) is the number of perfect matchings of an edge-labeled 2 X (2n+1) Mobius grid graph.
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
Offset: 0
Examples
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).
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).
Programs
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Magma
m:=25; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!((3-6*x+x^2)/((1-x)*(x^2-3*x+1)))); // G. C. Greubel, Sep 22 2018 -
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 *) -
PARI
{a(n) = 2 + fibonacci(2*n) + fibonacci(2*n+2)}; /* Michael Somos, Nov 03 2016 */
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
Comments