A153415 - OEIS

%I #36 Jan 14 2023 13:24:03

%S 5,6,6,1,7,7,6,7,5,8,1,1,3,8,4,5,5,0,2,7,5,9,2,9,3,2,1,2,1,2,0,6,2,0,

%T 0,3,7,3,6,1,4,4,1,9,7,8,6,5,9,0,5,5,7,0,4,9,2,3,4,4,4,1,3,2,5,4,5,7,

%U 5,5,5,4,5,3,0,2,0,8,6,8,5,6,1,4,8,5,5,6,7,8,4,2,1,8,1,8,3,2,6,6,4,6,1,5,3

%N Decimal expansion of Sum_{n>=1} 1/A000032(2*n).

%C From _Peter Bala_, Oct 15 2019: (Start)

%C c = (1/4)*(theta_3( (3-sqrt(5))/2 )^2 - 1 ), where theta_3(q) = 1 + 2*Sum_{n >= 1} q^n^2. See Borwein and Borwein, Proposition 3.5 (i), p. 91. Cf. A056854.

%C Series acceleration formulas (L(n) = A000032(n)):

%C c = 1 - 5*Sum_{n >= 1} 1/( L(2*n)*(L(2*n)^2 - 5) ).

%C c = (1/6) + 15*Sum_{n >= 1} 1/( L(2*n)*(L(2*n)^2 + 5) ).

%C c = (11/16) - 10*Sum_{n >= 1} (L(2*n)^2 - 10)/( L(2*n)*(L(2*n)^2 - 5)*(L(2*n)^2 - 20) ). (End)

%C Compare with Sum_{n >= 1} 1/(L(2*n) - sqrt(5)) = phi and Sum_{n >= 1} 1/(L(2*n) + sqrt(5)) = 2 - phi, where phi = (sqrt(5) + 1)/2. - _Peter Bala_, Nov 23 2019

%C This constant is transcendental (Duverney et al., 1997). - _Amiram Eldar_, Oct 30 2020

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 91.

%H Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, <a href="http://doi.org/10.3792/pjaa.73.140">Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers</a>, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.56617767581138455027...

%t First[ RealDigits[ N[(EllipticTheta[3, 0, GoldenRatio^(-2)]^2 - 1)/4, 120], 10, 105]](* _Jean-François Alcover_, Jun 07 2012, after _Eric W. Weisstein_ *)

%o (PARI) th3(x)=1 + 2*suminf(n=1,x^n^2)

%o phi=(sqrt(5)+1)/2

%o (th3(phi^-2)^2-1)/4 \\ _Charles R Greathouse IV_, Jun 06 2016

%Y Cf. A000032, A000122, A056854, A093540, A153416.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Dec 25 2008