A083332 a(n) = 10*a(n-2) - 16*a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.

1, 5, 14, 34, 124, 260, 1016, 2056, 8176, 16400, 65504, 131104, 524224, 1048640, 4194176, 8388736, 33554176, 67109120, 268434944, 536871424, 2147482624, 4294968320, 17179867136, 34359740416, 137438949376, 274877911040

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

Keywords

• nonn
• easy

Comments

a(n)/A083333(n) converges to 3.

Links

• Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -16).

Crossrefs

Cf. A147590, A081342 (bisections). [R. J. Mathar, Jul 13 2009]

Cf. A199710. [Bruno Berselli, Nov 11 2011]

Programs

Mathematica

CoefficientList[Series[(1+5x+4x^2-16x^3)/(1-10x^2+16x^4), {x, 0, 30}], x]

Maxima

(a[0] : 1, a[1] : 5, a[2] : 14, a[3] : 34, a[n] := 10*a[n - 2] - 16*a[n - 4], makelist(a[n], n, 0, 50));/* Franck Maminirina Ramaharo, Nov 12 2018 */

Formula

G.f.: (1 + 5*x + 4*x^2 - 16*x^3)/(1 - 10*x^2 + 16*x^4).

a(n) = A016116(n)*A014551(n+1). - R. J. Mathar, Jul 08 2009

From Franck Maminirina Ramaharo, Nov 12 2018: (Start)

a(n) = sqrt(2)^(3*n - 1)*(1 + sqrt(2) + (-1)^n*(sqrt(2) - 1)) + sqrt(2)^(n - 3)*(1 - sqrt(2) - (-1)^n*(sqrt(2) + 1)).

E.g.f.: (sinh(sqrt(2)*x) + 2*sinh(2*sqrt(2)*x))/sqrt(2) - cosh(sqrt(2)*x) + 2*cosh(2*sqrt(2)*x). (End)