A154997 - OEIS

%I #18 Dec 31 2023 11:36:08

%S 1,1,11,96,906,8316,77076,711936,6583896,60861456,562685616,

%T 5201957376,48092312736,444612597696,4110444968256,38001047740416,

%U 351319635490176,3247949245153536,30027284535626496,277602184568365056

%N a(n) = 6*a(n-1) + 30*a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

%C The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - _R. J. Mathar_, Jan 20 2009

%H G. C. Greubel, <a href="/A154997/b154997.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 (6,30).

%F G.f.: (1 -5*x -25*x^2)/(1 -6*x -30*x^2).

%F a(n+1) = Sum_{k=0..n} A154929(n,k)*5^(n-k).

%p m:=30; S:=series( (1-5*x-25*x^2)/(1-6*x-30*x^2), x, m+1):

%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Apr 21 2021

%t Join[{1},LinearRecurrence[{6,30},{1,11},20]] (* _Harvey P. Dale_, Feb 07 2012 *)

%o (Magma) I:=[1,11]; [1] cat [n le 2 select I[n] else 6*(Self(n-1) +5*Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Apr 21 2021

%o (Sage)

%o def A154996_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1-5*x-25*x^2)/(1-6*x-30*x^2) ).list()

%o A154996_list(30) # _G. C. Greubel_, Apr 21 2021

%Y Cf. A154996, A154999, A155000, A155001.

%K nonn

%O 0,3

%A _Philippe Deléham_, Jan 18 2009