A275860 a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = (3 + sqrt(13))/2, s = r/(r-1), c = 3, d = 1, a(0) = 1, a(1) = 1.

1, 1, 7, 33, 164, 813, 4039, 20063, 99665, 495099, 2459470, 12217747, 60693301, 301502133, 1497752387, 7440286381, 36960623072, 183606865105, 912091791531, 4530938620963, 22508046862781, 111811749387479, 555439900107962, 2759222392297991, 13706808258965257

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 (5,0,0,-4,0,1,-1).

Crossrefs

Cf. A098316, A275856, A275857, A275858, A275859, A275861.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-4*x+2*x^2-2*x^3+3*x^4-3*x^5+x^6)/(1-5*x+4*x^4-x^6+x^7))); // G. C. Greubel, Feb 08 2018

Mathematica

c = 3; 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+2*x^2-2*x^3+3*x^4-3*x^5+x^6)/(1-5*x+4*x^4-x^6+x^7), {x,0, 50}], x] (* G. C. Greubel, Feb 08 2018 *)
LinearRecurrence[{5,0,0,-4,0,1,-1},{1,1,7,33,164,813,4039},30] (* Harvey P. Dale, Sep 17 2024 *)

PARI

my(x='x+O('x^30)); Vec((1-4*x+2*x^2-2*x^3+3*x^4-3*x^5+x^6)/(1-5*x +4*x^4-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 = (3 + sqrt(13))/2 (A098316), s = r/(r-1), c = 3, d = 1, a(0) = 1, a(1) = 1.

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