OFFSET
0,2
COMMENTS
Inserting x=1/sqrt(b) into the power series expansion of arctanh(x) yields the general BBP-type formula log((sqrt(b)+1)/(sqrt(b)-1))*sqrt(b)/2 = Sum_{k>=0} 1/((2k+1)b^k).
This sequence illustrates case b=5, with
Sum_{k>=0} 1/a(k) = sqrt(5)*log((1+sqrt(5))/2).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
David H. Bailey, Compendium of BBP formulas for mathematical constants. See formula 86 at p. 26.
Index entries for linear recurrences with constant coefficients, signature (10,-25).
FORMULA
a(n) = 10*a(n-1) - 25*a(n-2).
O.g.f: (1+5*x)/(1-5*x)^2.
Sum_{n>=0} (-1)^n/a(n) = sqrt(5)*arctan(1/sqrt(5)). - Amiram Eldar, Feb 26 2022
E.g.f.: exp(5*x)*(1 + 10*x). - Stefano Spezia, May 09 2023
PROG
(PARI) a(n)=(2*n+1)*5^n
(Magma) [(2*n+1)*5^n: n in [0..25]]; // Vincenzo Librandi, Jun 08 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jaume Oliver Lafont, Dec 05 2009
STATUS
approved