1,1

A061037=0,5,3,21,2,45,15,77,6,117,35,: a(n)= A061037(12n+10)=(6n-1)*(6n+1)=36*n^2-1. From Balmer spectrum of hydrogen. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 25 2008]

If A=[A158589] 324*n.^2-18 (n>0, 306, 1278, 2898,.,); Y=[A005843] 2*n (n>0, 2, 4, 6,.,); X = [A136017] 36*n^2-1 (n>0, 35, 143, 323, .,), we have, for all terms, Pell's equation X^2-A*Y^2=1. Example: 35^2-306*2^2=1; 143^2-1278*4^2=1; 323^2-2898*6^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 22 2009]

Sum_{k>=1} (-1)^(k+1)/a(k) = (Pi-3)/6 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 20 2009]

Edward Everett Withford, Pell Equation [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 22 2009]

Vincenzo Librandi, X^2-AY^2=1 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 22 2009]

Wolfram MathWorld, Pell Equation [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 22 2009]

X. Gourdon and P. Sebah, Collection of series for Pi [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 20 2009]

O.g.f.: x*(-35-38*x+x^2)/(-1+x)^3 = 1-35/(-1+x)-108/(-1+x)^2-72/(-1+x)^3 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 12 2007

Table[36n^2 - 1, {n, 1, 100}]

(PARI) a(n)=36*n^2-1 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 20 2009]

Cf. A088878, A023208, A136016.

Cf. A005843, A158589 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 22 2009]

Sequence in context: A044748 A158586 A157286 this_sequence A048628 A048629 A133534

Adjacent sequences: A136014 A136015 A136016 this_sequence A136018 A136019 A136020

nonn

Artur Jasinski (grafix(AT)csl.pl), Dec 10 2007