|
%I #17 Jan 31 2021 11:29:33
%S 0,0,0,1,4,16,63,252,1008,4033,16132,64528,258111,1032444,4129776,
%T 16519105,66076420,264305680,1057222719,4228890876,16915563504,
%U 67662254017,270649016068,1082596064272,4330384257087,17321537028348
%N a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).
%H G. C. Greubel, <a href="/A135450/b135450.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-1,4).
%F a(n+1) - 4*a(n) = hexaperiodic 0, 0, 1, 0, 0, -1, A131531.
%F a(n) + a(n+3) = 1, 4, 16, 64 = 2^2n = A000302.
%F a(n) = (1/65)*4^n + (1/15)*(-1)^(n+1) + (2/39)*cos((Pi*n)/3) - (4*sqrt(3)/39) * sin((Pi*n)/3). Or, a(n) = (1/65)*(4^n + [ -1; -4; -16; 1; 4; 16]). - _Richard Choulet_, Dec 31 2007
%F O.g.f.: -x^3/[(4*x-1)*(1+x)*(x^2-x+1)]. - _R. J. Mathar_, Jan 07 2008
%t a = {0, 0, 0, 1, 4}; Do[AppendTo[a, 3*a[[ -1]] + 4*a[[ -2]] - a[[ -3]] + 3*a[[ -4]] + 4*a[[ -5]]], {25}]; a (* _Stefan Steinerberger_, Dec 31 2007 *)
%t LinearRecurrence[{3, 4, -1, 3, 4}, {0, 0, 0, 1, 4}, 25] (* _G. C. Greubel_, Oct 14 2016 *)
%t LinearRecurrence[{4,0,-1,4},{0,0,0,1},40] (* _Harvey P. Dale_, Jan 31 2021 *)
%o (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 4,-1,0,4]^n*[0;0;0;1])[1,1] \\ _Charles R Greathouse IV_, Oct 14 2016
%Y Cf. A135343, A135345.
%K nonn,easy
%O 0,5
%A _Paul Curtz_, Dec 14 2007
%E More terms from _Stefan Steinerberger_, Dec 31 2007
|
