login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = ((1+sqrt(3))*(5+sqrt(3))^n + (1-sqrt(3))*(5-sqrt(3))^n)/2.
4

%I #16 Dec 31 2022 15:17:58

%S 1,8,58,404,2764,18752,126712,854576,5758096,38780288,261124768,

%T 1758081344,11836068544,79682895872,536435450752,3611330798336,

%U 24311728066816,163668003104768,1101822013577728,7417524067472384,49935156376013824,336166034275745792,2263086902485153792

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

%C Fifth binomial transform of A162436, binomial transform of A161728.

%H Vincenzo Librandi, <a href="/A162272/b162272.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-22).

%F From _Emeric Deutsch_, Jul 05 2009: (Start)

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

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

%F E.g.f.: exp(5*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - _Stefano Spezia_, Dec 31 2022

%p 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

%t LinearRecurrence[{10, -22}, {1, 8}, 40] (* _Vincenzo Librandi_, Feb 03 2018 *)

%o (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

%Y Cf. A162436, A161728.

%K nonn,easy

%O 0,2

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

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 05 2009

%E Extended by _Emeric Deutsch_, Jul 05 2009