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%I #51 Jul 01 2023 15:18:57
%S 7,8,0,2,1,1,7,8,9,5,9,3,8,5,2,1,4,9,5,8,3,9,7,9,4,3,4,2,5,2,8,0,3,8,
%T 6,7,0,4,3,3,6,7,0,6,0,9,4,3,7,4,5,8,7,4,6,0,6,1,6,6,6,5,0,6,7,1,0,8,
%U 9,9,9,3,3,5,6,9,6,1,8,7,6,1,9,9,9,0,4,7,7,8,5,0,4,1,3,9,7,2,5,0,8
%N Decimal expansion of (phi^phi - 1/phi)/2, where phi = (1 + sqrt(5))/2 = A001622.
%C Conjecture: (phi^phi - 1/phi)/2 = integral_{x=0..oo} exp(-x*(2 + phi)) * (cosh(x) + sqrt(5) * sinh(x))^phi dx.
%H Alexander R. Povolotsky, <a href="https://groups.google.com/forum/#!topic/sci.math.research/AfEAu7rpcY8">Interesting Integral</a>, post in newsgroup sci.math.research, April 04, 2017.
%e 0.78021178959385214958397943425280386704336706094374587460616665067108999335696...
%t RealDigits[(GoldenRatio^GoldenRatio - 1/GoldenRatio)/2, 10, 120][[1]]
%o (Magma) (((Sqrt(5)+1)/2)^((Sqrt(5)+1)/2) - 1/((Sqrt(5)+1)/2))/2
%Y Cf. A001622 (golden ratio phi).
%K nonn,cons
%O 1,1
%A _Vincenzo Librandi_, Dec 29 2019
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