A154920 Denominators of a ternary BBP-type formula for log(3).

1, 18, 27, 324, 405, 4374, 5103, 52488, 59049, 590490, 649539, 6377292, 6908733, 66961566, 71744535, 688747536, 731794257, 6973568802, 7360989291, 69735688020, 73222472421, 690383311398, 721764371007, 6778308875544

Offset

0,2

Comments

log(3) = Sum_{k>=0} (9/(2k+1)+1/(2k+2))/9^(k+1).

log(3) = 1 + Sum_{k>=0} (1/(2k+2)+1/(2k+3))/9^(k+1).

Links

Table of n, a(n) for n=0..23.

David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants, 2017, page 14. [From Jaume Oliver Lafont, Sep 25 2009]

Index entries for linear recurrences with constant coefficients, signature (0,18,0,-81).

Formula

a(n) = (n+1)*9^[(n+1)/2] = 18*a(n-2) - 81*a(n-4).

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

G.f.: (1+18*x+9*x^2)/(1-9*x^2)^2. - Jaume Oliver Lafont, Jan 29 2009

a(n) = (2-(-1)^n)*(n+1)*3^n. - Jaume Oliver Lafont, Sep 27 2009

Sum_{n>=0} (-1)^n/a(n) = log(8/3). - Amiram Eldar, Feb 26 2022

Mathematica

LinearRecurrence[{0, 18, 0, -81}, {1, 18, 27, 324}, 30] (* Harvey P. Dale, Jan 10 2017 *)

Prog

(PARI) a(n)=(n+1)*9^((n+1)\2) \\ Jaume Oliver Lafont, Mar 25 2009
(Magma) [(2-(-1)^n)*(n+1)*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2015

Crossrefs

Cf. A002391, A058962.

Cf. A164985, A165132. - Jaume Oliver Lafont, Sep 25 2009

Sequence in context: A138336 A349485 A166630 * A094224 A128858 A141782

Adjacent sequences: A154917 A154918 A154919 * A154921 A154922 A154923

Keyword

nonn

Author

Jaume Oliver Lafont, Jan 17 2009, Jan 18 2009

Status

approved