OFFSET
1,1
COMMENTS
Extended to a(1)-a(2) using the formula/recurrence.
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 Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).
FORMULA
a(n) = 2^n + Lucas(2*n).
G.f.: x*(5 - 14*x + 6*x^2)/((1 - 2*x)*(1 - 3*x + x^2)). - Ilya Gutkovskiy, May 23 2017
From Colin Barker, Jun 05 2017: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>3.
a(n) = 2^(-n)*(4^n + (3-sqrt(5))^n + (3+sqrt(5))^n).
(End)
MATHEMATICA
Table[2^n + LucasL[2 n], {n, 20}]
LinearRecurrence[{5, -7, 2}, {5, 11, 26}, 20]
CoefficientList[Series[(-5 + 14 x - 6 x^2)/(-1 + 5 x - 7 x^2 + 2 x^3), {x, 0, 20}], x]
PROG
(Python)
from sympy import lucas
def a(n): return 2**n + lucas(2*n) # Indranil Ghosh, May 24 2017
(PARI) Vec(x*(5 - 14*x + 6*x^2)/((1 - 2*x)*(1 - 3*x + x^2)) + O(x^30)) \\ Colin Barker, Jun 05 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 23 2017
STATUS
approved