A276265 Expansion of (1 + 2*x)/(1 - 6*x + 6*x^2).

1, 8, 42, 204, 972, 4608, 21816, 103248, 488592, 2312064, 10940832, 51772608, 244990656, 1159308288, 5485905792, 25959585024, 122842075392, 581294942208, 2750717200896, 13016533552128, 61594898107392, 291470187331584, 1379251735345152, 6526689288081408, 30884625316417536

Offset: 0

Author

Ilya Gutkovskiy, Aug 26 2016

Keywords

• nonn
• easy

Comments

Satisfies recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=1, b(0)=2.

More generally, for the recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=k, b(0)=m solution is a(n) = (((sqrt(3) - 1)*k - 2*m)*(3 - sqrt(3))^n + (sqrt(3)*k + k + 2*m)*(3 + sqrt(3))^n)/(2*sqrt(3)), b(n) = ((-k + sqrt(3)*m + m)*(3 - sqrt(3))^n + (k + (sqrt(3) - 1)*m)*(3 + sqrt(3))^n)/(2*sqrt(3)).

Convolution of A030192 and {1, 2, 0, 0, 0, 0, 0, ...}.

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-6)

Crossrefs

Cf. A030192, A165663.

Programs

Maple

a:=series((1+2*x)/(1-6*x+6*x^2),x=0,25): seq(coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019

Mathematica

LinearRecurrence[{6, -6}, {1, 8}, 25]
CoefficientList[Series[(1 + 2 x)/(1 - 6 x + 6 x^2), {x, 0, 24}], x] (* Michael De Vlieger, Aug 26 2016 *)

PARI

Vec((1+2*x)/(1-6*x+6*x^2) + O(x^99)) \\ Altug Alkan, Aug 26 2016

Formula

O.g.f.: (1 + 2*x)/(1 - 6*x + 6*x^2).

E.g.f.: (5*sqrt(3)*sinh(sqrt(3)*x) + 3*cosh(sqrt(3)*x))*exp(3*x)/3.

a(n) = 6*a(n-1) - 6*a(n-2).

a(n) = ((-5 + sqrt(3))*(3 - sqrt(3))^n + (5 + sqrt(3))*(3 + sqrt(3))^n)/(2*sqrt(3)).

Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(3) = A165663.

a(n) = A030192(n)+2*A030192(n-1). - R. J. Mathar, Jan 25 2023