A275859 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.

1, 1, 5, 19, 76, 304, 1220, 4898, 19667, 78971, 317103, 1273309, 5112902, 20530578, 82439414, 331030964, 1329236757, 5337477605, 21432349833, 86060430295, 345570957936, 1387621309348, 5571917587224, 22373730779190, 89840494074695, 360749597608127

Offset

0,3

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-4,0,1,-1).

Formula

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

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

Mathematica

c = 2; 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[(1-4*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 *)
LinearRecurrence[{5, -4, 0, 1, -1}, {1, 1, 5, 19, 76}, 30] (* Harvey P. Dale, Apr 23 2019 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A149769 A149770 A285424 * A323269 A268815 A108981

Adjacent sequences: A275856 A275857 A275858 * A275860 A275861 A275862

Keyword

nonn,easy

Author

Clark Kimberling, Aug 12 2016

Status

approved