A022314 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0, a(1) = 9.

0, 9, 10, 20, 31, 52, 84, 137, 222, 360, 583, 944, 1528, 2473, 4002, 6476, 10479, 16956, 27436, 44393, 71830, 116224, 188055, 304280, 492336, 796617, 1288954, 2085572, 3374527, 5460100, 8834628, 14294729, 23129358, 37424088, 60553447, 97977536, 158530984

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = 9*x + 10*x^2 + 20*x^3 + 31*x^4 + 52*x^5 + 84*x^6 + 137*x^7 + 222*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A022100.

Programs

Mathematica

LinearRecurrence[{2, 0, -1}, {0, 9, 10}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
a[ n_] := 9 Fibonacci[n] + Fibonacci[n + 1] - 1; (* Michael Somos, Nov 21 2016 *)

PARI

concat(0, Vec(-x*(-9+8*x) / ( (x-1)*(x^2+x-1) ) + O(x^30))) \\ Michel Marcus, Nov 20 2016
{a(n) = 9*fibonacci(n) + fibonacci(n+1) - 1}; /* Michael Somos, Nov 21 2016 */

Formula

a(n) = -1 + (1/2)*((1 + sqrt(5))/2)^n + (19/10)sqrt(5)*((1 + sqrt(5))/2)^n - (19/10)*sqrt(5)*((1 - sqrt(5))/2)^n + (1/2)*((1 - sqrt(5))/2)^n, obtained using PURRS. - Alexander R. Povolotsky, Apr 22 2008

From R. J. Mathar, Apr 07 2011: (Start)

G.f.: -x*(-9+8*x) / ( (x-1)*(x^2+x-1) ).

a(n) = A022100(n) - 1. (End)

a(n) = F(n+2) + 8*F(n) - 1, where A000045. - G. C. Greubel, Aug 25 2017