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%I #20 Sep 08 2022 08:45:48
%S 0,0,1,4,25,100,481,1924,8425,33700,141361,565444,2320825,9283300,
%T 37664641,150658564,607417225,2429668900,9761722321,39046889284,
%U 156574977625,626299910500,2508686426401,10034745705604,40170363882025
%N a(n) = (-7*3^n+(-3)^n+6*4^n) / 42.
%C This is the main sequence for the recurrence a(n) = 4*a(n-1) + 9*a(n-2) - 36*a(n-3) in the sense that the numerator of the g.f. contains only a power of x.
%H G. C. Greubel, <a href="/A167889/b167889.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,9,-36).
%F a(n+1) - 4*a(n) = 0 if n even, = A001019((n-1)/2) if n odd.
%F G.f. x^2/( (3*x-1)*(3*x+1)*(4*x-1) ). - _R. J. Mathar_, Jul 04 2011
%F From _G. C. Greubel_, Jun 30 2016: (Start)
%F a(n) = 4*a(n-1) + 9*a(n-2) - 36*a(n-3).
%F E.g.f.: (1/42)*(exp(-3*x) - 7*exp(3*x) + 6*exp(4 x)). (End)
%t LinearRecurrence[{4,9,-36},{0,0,1}, 100] (* or *) Table[(-7*3^n + (-3)^n + 6*4^n)/42, {n, 0, 50}] (* _G. C. Greubel_, Jun 30 2016 *)
%o (Magma) [(-7*3^n+(-3)^n+6*4^n) / 42: n in [0..30]]; // _Vincenzo Librandi_, Jul 01 2016
%K nonn,easy
%O 0,4
%A _Paul Curtz_, Nov 14 2009
%E More terms from _Vincenzo Librandi_, Apr 05 2010
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