A367120 Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).
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, 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, 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
Offset: 1
Examples
2.224412437956340467163837541384021939...
Programs
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Mathematica
First[RealDigits[2/HypergeometricPFQ[{1, 1}, {3/2, 3}, -1/2], 10, 100]] (* or *) First[RealDigits[2/Sum[(-1)^k/Binomial[k+2, 2]/(2*k+1)!!, {k, 0, Infinity}], 10, 100]] (* Paolo Xausa, Nov 18 2024 *) -
PARI
N=50; doblfac(n) = if(n<0, 0, n<2, 1, n*doblfac(n-2)); ap1 = 2 / sum(k=0,N, (-1)^k/binomial(k+2,2)/doblfac(2*k+1)); ap2 = 2 / sum(k=0,N+1, (-1)^k/binomial(k+2,2)/doblfac(2*k+1)); n = 0; while(digits(floor(10^(n+1)*ap1)) == digits(floor(10^(n+1)*ap2)), n++); A367120 = digits(floor(10^n*ap1));
Formula
Equals 2 / pFq(1,1; 3/2,3; -1/2) where pFq() is the generalized hypergeometric function.
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 + ... ).