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%I #31 May 09 2023 19:46:33
%S 1,15,125,875,5625,34375,203125,1171875,6640625,37109375,205078125,
%T 1123046875,6103515625,32958984375,177001953125,946044921875,
%U 5035400390625,26702880859375,141143798828125,743865966796875,3910064697265625,20503997802734375,107288360595703125
%N a(n) = (2n + 1)*5^n.
%C 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).
%C This sequence illustrates case b=5, with
%C Sum_{k>=0} 1/a(k) = sqrt(5)*log((1+sqrt(5))/2).
%H Vincenzo Librandi, <a href="/A171220/b171220.txt">Table of n, a(n) for n = 0..200</a>
%H David H. Bailey, <a href="https://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf">Compendium of BBP formulas for mathematical constants</a>. See formula 86 at p. 26.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-25).
%F a(n) = 10*a(n-1) - 25*a(n-2).
%F O.g.f: (1+5*x)/(1-5*x)^2.
%F Sum_{n>=0} (-1)^n/a(n) = sqrt(5)*arctan(1/sqrt(5)). - _Amiram Eldar_, Feb 26 2022
%F E.g.f.: exp(5*x)*(1 + 10*x). - _Stefano Spezia_, May 09 2023
%o (PARI) a(n)=(2*n+1)*5^n
%o (Magma) [(2*n+1)*5^n: n in [0..25]]; // _Vincenzo Librandi_, Jun 08 2011
%Y 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).
%K nonn,easy
%O 0,2
%A _Jaume Oliver Lafont_, Dec 05 2009
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