A275856 a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = (1+sqrt(5))/2, s = r/(r-1), c = 1, d = 1, a(0) = 1, a(1) = 1.

1, 1, 3, 8, 24, 74, 231, 723, 2265, 7098, 22246, 69724, 218533, 684941, 2146791, 6728628, 21089356, 66099798, 207174811, 649342415, 2035216397, 6378923798, 19993288618, 62664424664, 196407414185, 615594454969, 1929441078187, 6047395073408, 18954187089376

Offset: 0

Author

Clark Kimberling, Aug 11 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 (4,-3,1,0,-1).

Crossrefs

Cf. A275857-A275861.

Programs

Mathematica

c = 1; 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}]
LinearRecurrence[{4,-3,1,0,-1},{1,1,3,8,24},40] (* Harvey P. Dale, Mar 21 2025 *)

Formula

a(n) = floor(s*a(n-1)) + floor(r*a(n-2)), where r = (1+sqrt(5))/2, s = r/(r-1).

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