A119032 a(n+2) = 18*a(n+1) - a(n) + 8.

0, 9, 170, 3059, 54900, 985149, 17677790, 317215079, 5692193640, 102142270449, 1832868674450, 32889493869659, 590178020979420, 10590314883759909, 190035489886698950, 3410048503076821199, 61190837565496082640, 1098025027675852666329, 19703259660599851911290

Offset: 1

Author

Richard Choulet, Aug 30 2007, Oct 09 2007

Keywords

• nonn
• easy

Comments

Arises in calculating A107075. A053606 follows the same recurrence.

Links

• Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Crossrefs

Cf. A000032, A020837, A053606, A107075.

Programs

Mathematica

LinearRecurrence[{19, -19, 1}, {0, 9, 170}, 20] (* Amiram Eldar, Dec 02 2024 *)

Formula

a(n+1) = 9*a(n+1) + 4 + (80*a(n)^2+80*a(n)+25)^(1/2).

G.f.: (9*x-x^2)/((1-x)*(1-18*x+x^2)).

a(n) = ((sqrt(5)+2)/8)*(9+4*sqrt(5))^(n-1) + ((-sqrt(5)+2)/8)*(9-4*sqrt(5))^(n-1) - 1/2. - Richard Choulet, Nov 26 2008

a(n) = (Lucas(6*n-3)-4)/8, where Lucas(n) = A000032(n). - Gary Detlefs, Dec 07 2010

Product_{n>=2} (1 + 1/a(n)) = sqrt(5)/2 (= 10 * A020837). - Amiram Eldar, Dec 02 2024