A162272 a(n) = ((1+sqrt(3))*(5+sqrt(3))^n + (1-sqrt(3))*(5-sqrt(3))^n)/2.

1, 8, 58, 404, 2764, 18752, 126712, 854576, 5758096, 38780288, 261124768, 1758081344, 11836068544, 79682895872, 536435450752, 3611330798336, 24311728066816, 163668003104768, 1101822013577728, 7417524067472384, 49935156376013824, 336166034275745792, 2263086902485153792

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

• nonn
• easy

Comments

Fifth binomial transform of A162436, binomial transform of A161728.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-22).

Crossrefs

Cf. A162436, A161728.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((1+r)*(5+r)^n+(1-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009

Maple

seq(expand(((1+sqrt(3))*(5+sqrt(3))^n+(1-sqrt(3))*(5-sqrt(3))^n)*1/2), n = 0 .. 20); # Emeric Deutsch, Jul 05 2009

Mathematica

LinearRecurrence[{10, -22}, {1, 8}, 40] (* Vincenzo Librandi, Feb 03 2018 *)

Formula

From Emeric Deutsch, Jul 05 2009: (Start)

G.f.: (1 - 2*x)/(1 - 10*x + 22*x^2).

a(n) = 10*a(n-1) - 22*a(n-2) for n >= 2; a(0)=1, a(1)=8. (End)

E.g.f.: exp(5*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 05 2009

Extended by Emeric Deutsch, Jul 05 2009