A275861 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.

1, 1, 9, 51, 305, 1813, 10784, 64144, 381543, 2269503, 13499513, 80298135, 477631347, 2841058559, 16899254596, 100520563016, 597918892325, 3556555903317, 21155193548465, 125835844069155, 748499871500621, 4452245397810113, 26482955892270832

Offset: 0

Author

Clark Kimberling, Aug 12 2016

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-7,5,-3,-3,3,-1).

Crossrefs

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

Programs

Magma

Q:=Rationals(); R:=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

Mathematica

c = 4; d = 1; z = 40;
r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;
a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];
t = Table[a[n], {n, 0, z}]
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 *)
LinearRecurrence[{7,-7,5,-3,-3,3,-1},{1,1,9,51,305,1813,10784},40] (* Harvey P. Dale, Dec 21 2018 *)

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

Formula

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.

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).