A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686

Offset: 0

Author

Paul Barry, Nov 15 2003

Keywords

• easy
• nonn
• changed

Comments

The family of recurrences a(n) = 2*k*a(n-1) + a(n-2), a(0)=1, a(1)=k has solution a(n) = ((k+sqrt(k^2+1))^n + (k-sqrt(k^2+1))^n)/2; a(n) = Sum_{j=0..floor(n/2)} C(n,2k)*(k^2+1)^jk^(n-2j); a(n) = T(n,ki)*(-i)^n; e.g.f. exp(kx)*cosh(sqrt(k^2+1)*x).

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (12,1).

Crossrefs

Cf. A088320, A088317, A005667, A001077.

Essentially the same as A041060.

Formula

E.g.f.: exp(6x)*cosh(sqrt(37)x);

a(n) = ((6+sqrt(37))^n + (6-sqrt(37))^n)/2;

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*37^k*6^(n-2k).

a(n) = T(n, 6i)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.

G.f.: (1-6x)/(1-12*x-x^2). - Philippe Deléham, Nov 21 2008