0,3

a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=3, beta =1 (or alpha=1, beta=3).

S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 39, equation (50).

G.f.: (1+3*x*c(3*x)/2)/(1+x/2) = 1/(1-x*c(3*x)) with c(x) g.f. of Catalan numbers A000108.

a(n)= sum((n-m)*binomial(n-1+m, m)*(3^m)/n, m=0..n-1) = ((-1/2)^n)*(1-3*sum(C(k)*(-6)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum{ k= 0...n, A059365(n, k)*3^(n-k) }. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 19 2004

Given the semi-axes a,b of an ellipse, then Ramanujan gave the highly accurate formula for the perimeter p = pi((a+b) + (3(a-b)^2)/ (10(a+b) +sqrt(a^2 +14ab +b^2))). If we let h = ((a-b)/ (a+b))^2, then (p/(pi (a+b)) -1)/4 = (3/4)* h/(10 +sqrt( 4 -3*h )) = 1*(h/16) +1*(h/16)^2 +4*(h/16)^3 +25*(h/16)^4 +... - Michael Somos Apr 11 2007

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-2*x^2)/(1+x)^2+O(x^(n+1))), n)) (from R. Stephan)

(PARI) {a(n)= if(n<1, n==0, polcoeff( serreverse( (x-2*x^2)/ (1+x)^2 +x*O(x^n)), n))} /* Michael Somos Apr 11 2007 */

A064062 (C(2, n)).

Sequence in context: A051820 A054368 A135147 this_sequence A060908 A036449 A051500

Adjacent sequences: A064060 A064061 A064062 this_sequence A064064 A064065 A064066

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 2001