A166725 -id:A166725 - cp's OEIS frontend

1, 15, 125, 875, 5625, 34375, 203125, 1171875, 6640625, 37109375, 205078125, 1123046875, 6103515625, 32958984375, 177001953125, 946044921875, 5035400390625, 26702880859375, 141143798828125, 743865966796875, 3910064697265625, 20503997802734375, 107288360595703125

Offset: 0

Jaume Oliver Lafont, Dec 05 2009

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).

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).

Cf. A014480 ((2n+1)*2^n), A124647 ((2n+1)*3^n), A058962 ((2n+1)*4^n), A155988 ((2n+1)*9^n), A165283 ((2n+1)*16^n), A166725 ((2n+1)*25^n).

[(2*n+1)*5^n: n in [0..25]]; // Vincenzo Librandi, Jun 08 2011

PARI
```
a(n)=(2*n+1)*5^n
```

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

cp's OEIS Frontend

A171220 a(n) = (2n + 1)*5^n.

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