A072264 - OEIS

%I #25 Dec 31 2023 10:25:16

%S 1,1,8,29,127,526,2213,9269,38872,162961,683243,2864534,12009817,

%T 50352121,211105448,885076949,3710758087,15557659006,65226767453,

%U 273468597389,1146539629432,4806961875241,20153583772883,84495560694854,354254600948977,1485241606321201

%N a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=1.

%H G. C. Greubel, <a href="/A072264/b072264.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 (3,5).

%F G.f.: (1-2*x)/(1-3*x-5*x^2). - _Jaume Oliver Lafont_, Mar 06 2009

%F G.f.: G(0)*(1-2*x)/(2-3*x), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 17 2013

%F a(n) = 5^((n-1)/2)*( sqrt(5)*Fibonacci(n+1, 3/sqrt(5)) - 2*Fibonacci(n, 3/sqrt(5)) ). - _G. C. Greubel_, Jan 14 2020

%e a(5)=3*a(4)+5*a(3): 127=3*29+5*8=87+40.

%p seq(coeff(series((1-2*x)/(1-3*x-5*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Jan 14 2020

%t LinearRecurrence[{3,5},{1,1},30] (* _Harvey P. Dale_, Feb 17 2018 *)

%o (Magma) [n le 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..26]];  // _Bruno Berselli_, Oct 11 2011

%o (PARI) my(x='x+O('x^30)); Vec((1-2*x)/(1-3*x-5*x^2)) \\ _G. C. Greubel_, Jan 14 2020

%o (Sage)

%o def A072264_list(prec):

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

%o     return P( (1-2*x)/(1-3*x-5*x^2) ).list()

%o A072264_list(30) # _G. C. Greubel_, Jan 14 2020

%o (GAP) a:=[1,1];; for n in [3..30] do a[n]:=3*a[n-1]+5*a[n-2]; od; a; # _G. C. Greubel_, Jan 14 2020

%Y Cf. A015523, A072263, A152187, A179606, A197189.

%K nonn,easy

%O 0,3

%A _Miklos Kristof_, Jul 08 2002

%E Offset changed and more terms added by _Bruno Berselli_, Oct 11 2011