A275859 - OEIS

%I #17 Sep 08 2022 08:46:17

%S 1,1,5,19,76,304,1220,4898,19667,78971,317103,1273309,5112902,

%T 20530578,82439414,331030964,1329236757,5337477605,21432349833,

%U 86060430295,345570957936,1387621309348,5571917587224,22373730779190,89840494074695,360749597608127

%N a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = 1 + sqrt(2), s = r/(r-1), c = 2, d = 1, a(0) = 1, a(1) = 1.

%H Clark Kimberling, <a href="/A275859/b275859.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,0,1,-1).

%F a(n) = floor(s*a(n-1)) + floor(r*a(n-2)), where r = 1 + sqrt(2), s = r/(r-1).

%F G.f.: (1 - 4*x + 4*x^2 - 2*x^3)/(1 - 5*x + 4*x^2 - x^4 + x^5).

%t c = 2; d = 1; z = 40;

%t r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;

%t a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];

%t t = Table[a[n], {n, 0, z}]

%t CoefficientList[Series[(1-4*x+4*x^2-2*x^3)/(1-5*x+4*x^2-x^4+x^5), {x,0, 50}], x] (* _G. C. Greubel_, Feb 08 2018 *)

%t LinearRecurrence[{5,-4,0,1,-1},{1,1,5,19,76},30] (* _Harvey P. Dale_, Apr 23 2019 *)

%o (PARI) x='x+O('x^30); Vec((1-4*x+4*x^2-2*x^3)/(1-5*x+4*x^2-x^4+x^5)) \\ _G. C. Greubel_, Feb 08 2018

%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-4*x+4*x^2-2*x^3)/(1-5*x+4*x^2-x^4+x^5))) // _G. C. Greubel_, Feb 08 2018

%Y Cf. A275856, A275857, A275858, A275860, A275861.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Aug 12 2016