A287349 Number of matchings in the n-gear graph.

4, 13, 42, 131, 398, 1186, 3482, 10103, 29034, 82777, 234424, 660098, 1849552, 5160001, 14341098, 39723791, 109701122, 302131618, 830079014, 2275509227, 6225274794, 16999389733, 46341292012, 126130604546, 342800478748, 930414584821, 2522124577962, 6828859302683

Offset

1,1

Links

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

Eric Weisstein's World of Mathematics, Gear Graph

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Matching

Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Formula

a(n) = Lucas(2*n) + n*Fibonacci(2*n) for n > 0.

G.f.: x*(4 - 11*x + 8*x^2 - 2*x^3)/(1 - 3*x + x^2)^2. - Ilya Gutkovskiy, May 23 2017

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n>4. - Colin Barker, Jun 05 2017

Mathematica

Table[LucasL[2 n] + n Fibonacci[2 n], {n, 20}]
LinearRecurrence[{6, -11, 6, -1}, {4, 13, 42, 131}, 30]
CoefficientList[Series[(42 - 121 x + 74 x^2 - 13 x^3)/(1 - 3 x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Oct 02 2017 *)

Prog

(Python)
from sympy import lucas, fibonacci
def a(n): return lucas(2*n) + n*fibonacci(2*n) # Indranil Ghosh, May 24 2017
(PARI) Vec(x*(4 - 11*x + 8*x^2 - 2*x^3)/(1 - 3*x + x^2)^2 + O(x^30)) \\ Colin Barker, Jun 05 2017
(PARI) a(n) = fibonacci(2*n-1) + n*fibonacci(2*n) + fibonacci(2*n+1); \\ Altug Alkan, Oct 02 2017

Crossrefs

Cf. A000032, A000045.

Sequence in context: A109454 A357063 A307261 * A000640 A199842 A192910

Adjacent sequences: A287346 A287347 A287348 * A287350 A287351 A287352

Keyword

nonn,easy

Author

Eric W. Weisstein, May 23 2017

Status

approved