A367120 - OEIS

%I #31 Nov 18 2024 03:37:55

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

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

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

%N Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).

%F Equals 2 / pFq(1,1; 3/2,3; -1/2) where pFq() is the generalized hypergeometric function.

%F Equals 2 / Sum_{k>=0} (-1)^k/binomial(k+2,2)/(2*k+1)!! = 2 / (1 - 1/9 + 1/90 - 1/1050 + 1/14175 - 1/218295 + ... ).

%e 2.224412437956340467163837541384021939...

%t First[RealDigits[2/HypergeometricPFQ[{1, 1}, {3/2, 3}, -1/2], 10, 100]] (* or *)

%t First[RealDigits[2/Sum[(-1)^k/Binomial[k+2, 2]/(2*k+1)!!, {k, 0, Infinity}], 10, 100]] (* _Paolo Xausa_, Nov 18 2024 *)

%o (PARI)

%o N=50;

%o doblfac(n) = if(n<0, 0, n<2, 1, n*doblfac(n-2));

%o ap1 = 2 / sum(k=0,N, (-1)^k/binomial(k+2,2)/doblfac(2*k+1));

%o ap2 = 2 / sum(k=0,N+1, (-1)^k/binomial(k+2,2)/doblfac(2*k+1));

%o n = 0; while(digits(floor(10^(n+1)*ap1)) == digits(floor(10^(n+1)*ap2)), n++);

%o A367120 = digits(floor(10^n*ap1));

%Y Cf. A113014, A113011, A194807, A365307.

%K nonn,cons

%O 1,1

%A _Rok Cestnik_, Nov 13 2023