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A287349
Number of matchings in the n-gear graph.
2
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
Eric Weisstein's World of Mathematics, Gear Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
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
Sequence in context: A109454 A357063 A307261 * A000640 A199842 A192910
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 23 2017
STATUS
approved