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

Author

Jaume Oliver Lafont, Jan 17 2009, Jan 18 2009

Keywords

• nonn

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

• 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).

Crossrefs

Cf. A002391, A058962.

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

Programs

Magma

[(2-(-1)^n)*(n+1)*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2015

Mathematica

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

PARI

a(n)=(n+1)*9^((n+1)\2) \\ Jaume Oliver Lafont, Mar 25 2009

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