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%I #8 Jul 03 2021 18:09:33
%S 2,5,6,2,7,9,3,5,3,4,7,8,3,1,8,9,4,6,1,6,0,7,6,8,1,6,4,5,1,3,8,5,7,1,
%T 3,3,5,1,5,0,8,4,9,0,6,7,8,9,2,0,6,6,1,1,9,2,2,7,8,9,6,7,9,2,8,8,8,6,
%U 8,5,3,8,7,0,8,5,7,6,4,5,8,4,9,7,2,5,5,4,1,2,4,2,5,3,1,7,5,9,5,9
%N Decimal expansion of Sum_{k>=0} 2^floor(k/2)/(k!^2).
%C This constant is irrational (Mingarelli, 2013).
%H Angelo B. Mingarelli, <a href="http://nntdm.net/volume-19-2013/number-4/43-76/">Abstract factorials</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, No. 4 (2013), pp. 43-76 (see p. 62); <a href="https://arxiv.org/abs/0705.4299">arXiv preprint</a>, arXiv:0705.4299 [math.NT], 2007-2012.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html">Bessel Function of the First Kind</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">Modified Bessel Function of the First Kind</a>.
%F Equals (1/4)*(2+sqrt(2)) * BesselI(0,2^(5/4)) + (1/4)*(2-sqrt(2)) * BesselJ(0, 2^(5/4)), where BesselJ is the Bessel function of the first kind, and BesselI is the modified Bessel function of the first kind.
%e 2.56279353478318946160768164513857133515084906789206...
%p evalf(sum(2^floor(k/2)/k!^2, k=0..infinity), 140); # _Alois P. Heinz_, Jul 03 2021
%t RealDigits[(1/4) * (2+Sqrt[2]) * BesselI[0,2^(5/4)] + (1/4) * (2-Sqrt[2]) * BesselJ[0,2^(5/4)], 10, 100][[1]]
%o (PARI) suminf(k=0, 2^floor(k/2)/(k!^2)) \\ _Michel Marcus_, Jul 02 2021
%Y Cf. A001044, A016116.
%K nonn,cons
%O 1,1
%A _Amiram Eldar_, Jul 02 2021
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