A275861 - OEIS

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

%S 1,1,9,51,305,1813,10784,64144,381543,2269503,13499513,80298135,

%T 477631347,2841058559,16899254596,100520563016,597918892325,

%U 3556555903317,21155193548465,125835844069155,748499871500621,4452245397810113,26482955892270832

%N a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = 2 + sqrt(5), s = r/(r-1), c = 4, d = 1, a(0) = 1, a(1) = 1.

%H Clark Kimberling, <a href="/A275861/b275861.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,5,-3,-3,3,-1).

%F a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = 2 + sqrt(5), s = r/(r-1), c = 4, d = 1, a(0) = 1, a(1) = 1.

%F G.f.: (1 -6*x +9*x^2 -10*x^3 +9*x^4 -4*x^5)/(1 -7*x +7*x^2 -5*x^3 +3*x^4 +3*x^5 -3*x^6 +x^7).

%t c = 4; 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[(1-6*x+9*x^2-10*x^3+9*x^4-4*x^5)/(1-7*x+7*x^2 -5*x^3+3*x^4+3*x^5-3*x^6+x^7), {x,0, 50}], x] (* _G. C. Greubel_, Feb 08 2018 *)

%t LinearRecurrence[{7,-7,5,-3,-3,3,-1},{1,1,9,51,305,1813,10784},40] (* _Harvey P. Dale_, Dec 21 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-6*x+9*x^2-10*x^3+9*x^4-4*x^5)/(1-7*x+7*x^2 -5*x^3+3*x^4+3*x^5-3*x^6+x^7)) \\ _G. C. Greubel_, Feb 08 2018

%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-6*x+9*x^2-10*x^3+9*x^4-4*x^5)/(1-7*x+7*x^2 -5*x^3+3*x^4+ 3*x^5- 3*x^6+x^7))) // _G. C. Greubel_, Feb 08 2018

%Y Cf. A275856, A275857, A275858, A275859, A275860.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Aug 12 2016