A155988 a(n) = (2*n + 1)*9^n.

1, 27, 405, 5103, 59049, 649539, 6908733, 71744535, 731794257, 7360989291, 73222472421, 721764371007, 7060738412025, 68630377364883, 663426981193869, 6382625094934119, 61149666232110753, 583701359488329915, 5553501505988967477, 52683216989246691471, 498464283821334080841

Offset: 0

Author

Jaume Oliver Lafont, Feb 01 2009

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants, 2017, page 14.
• Xavier Gourdon and Pascal Sebah, Collection of formulas for log 2.
• Index entries for linear recurrences with constant coefficients, signature (18,-81).

Crossrefs

Cf. A058962 for the similar (2n+1)4^n.

Cf. A001019, A005408, A060851, A096949, A096950, A154920, A164985, A165132.

Programs

Magma

[(2*n+1)*9^n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2011

Maxima

makelist((2*n+1)*9^n, n, 0, 20); /* Martin Ettl, Nov 11 2012 */

PARI

a(n)=(2*n+1)*9^n;

Formula

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

a(n) = 18*a(n-1) - 81*a(n-2) for n>=2.

Sum_{n>=0} 1/a(n) = (3/2)*log(2).

a(n) = A005408(n) * A001019(n).

a(n) = (2*n - 1)*3^(2*n-1)/3 = A060851(n)/3.

Sum_{n>=0} (-1)^n/a(n) = 3*arctan(1/3). - Amiram Eldar, Feb 26 2022

E.g.f.: exp(9*x)*(1 + 18*x). - Stefano Spezia, May 07 2023