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

0,2

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).

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

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 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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

Crossrefs

Cf. A162436, A161728.

Sequence in context: A297097 A081897 A125371 * A273584 A037532 A062236

Adjacent sequences: A162269 A162270 A162271 * A162273 A162274 A162275

Keyword

nonn,easy

Author

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

Extensions

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

Extended by Emeric Deutsch, Jul 05 2009

Status

approved