A088014 Expansion of e.g.f.: cosh(sqrt(2)*x)*(1+exp(x)).
2, 1, 5, 7, 21, 41, 107, 239, 593, 1393, 3395, 8119, 19665, 47321, 114371, 275807, 666113, 1607521, 3881411, 9369319, 22620561, 54608393, 131838371, 318281039, 768402497, 1855077841, 4478562275, 10812186007, 26102942481, 63018038201
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-2)
Programs
-
Magma
m:=50; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!((x-2)*(2*x-1)*(1+x)/((2*x^2-1)*(x^2+2*x-1)))); // G. C. Greubel, Aug 16 2018 -
Mathematica
With[{nn=30},CoefficientList[Series[Cosh[Sqrt[2]x](1+Exp[x]),{x,0,nn}],x]Range[0,nn]!] (* or *) LinearRecurrence[{2,3,-4,-2},{2,1,5,7},30] (* Harvey P. Dale, Jul 31 2012 *) -
PARI
x='x+O('x^50); Vec((x-2)*(2*x-1)*(1+x)/((2*x^2-1)*(x^2+2*x-1))) \\ G. C. Greubel, Aug 16 2018
Formula
G.f.: (x-2)*(2*x-1)*(1+x) / ( (2*x^2-1)*(x^2+2*x-1) ).
E.g.f.: cosh(sqrt(2)*x)*(1+exp(x)).
a(n) = ((sqrt(2))^n + (-sqrt(2))^n + (1+sqrt(2))^n + (1-sqrt(2))^n)/2.
a(0)=2, a(1)=1, a(2)=5, a(3)=7, a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 2*a(n-4). - Harvey P. Dale, Jul 31 2012