A161727 a(n) = ((2+sqrt(3))*(4+sqrt(3))^n-(2-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).

1, 6, 35, 202, 1161, 6662, 38203, 219018, 1255505, 7196806, 41252883, 236464586, 1355429209, 7769394054, 44534572715, 255274459018, 1463246226849, 8387401847558, 48077013831427, 275579886633162, 1579637913256745

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Fourth binomial transform of A038754, binomial transform of A140766.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-13).

Crossrefs

Cf. A038754, A140766.

Programs

Maple

seq(expand(((2+sqrt(3))*(4+sqrt(3))^n-(2-sqrt(3))*(4-sqrt(3))^n)/sqrt(12)), n = 0 .. 20) # Emeric Deutsch, Jun 20 2009

Mathematica

LinearRecurrence[{8,-13},{1,6},30] (* Harvey P. Dale, Jun 01 2016 *)

PARI

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

Formula

a(n) = 8*a(n-1)-13(n-2) for n > 1; a(0) = 1, a(1) = 6.

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

a(n) = A153594(n+1)-2*A153594(n). - R. J. Mathar, Feb 04 2021

Extensions

Extended beyond a(6) by Klaus Brockhaus and Emeric Deutsch, Jun 19 2009

Edited by Klaus Brockhaus, Jul 05 2009