A161728 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n-(3-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).

1, 7, 43, 253, 1465, 8431, 48403, 277621, 1591729, 9124759, 52305595, 299822893, 1718610409, 9851185663, 56467549987, 323674986277, 1855321740385, 10634799101479, 60959210186827, 349421293175389, 2002900612974361, 11480728092514831, 65808116771451955, 377215468968922837

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Keywords

• nonn
• easy

Comments

Fourth binomial transform of A162436, inverse binomial transform of A162272.

The inverse binomial transform yields A030192. The binomial transform yields A162272. - R. J. Mathar, Jul 07 2009

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-13).

Crossrefs

Cf. A162436, A162272, A030192, A161729.

Programs

PARI

F=nfinit(x^2-3); for(n=0, 20, print1(nfeltdiv(F, ((3+x)*(4+x)^n-(3-x)*(4-x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009

PARI

Vec((1-x)/(1-8*x+13*x^2)+O(x^25)) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = 8*a(n-1) - 13(n-2) for n > 1; a(0) = 1, a(1) = 7.

G.f.: (1 - x)/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Jun 19 2009

E.g.f.: exp(4*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Extensions

Extended beyond a(5) by Klaus Brockhaus, Jun 19 2009

Edited by Klaus Brockhaus, Jul 05 2009; M. F. Hasler, Dec 03 2014