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%I #26 Nov 23 2018 08:21:42
%S 2,0,8,5,1,2,4,1,1,7,6,3,4,3,9,3,7,2,3,8,0,3,3,6,8,6,0,5,9,7,5,1,0,4,
%T 9,2,6,4,6,6,4,4,9,8,4,9,1,7,0,0,5,6,0,3,9,9,1,6,6,8,2,0,4,7,5,6,8,5,
%U 4,5,9,4,7,2,6,8,3,3,8,0,6,0,8,6,3,3,6,8,5,7,2,8,4,7,5,3,9,1,6,6,6,2,3,2,0,2,9,6,0,5,2,3,7,8,3,3,9,6,8,5,8,7,9,2,3,4,5,6,2,0,5,2,3,1,1,2,1,1,7,2,9,3,5,5,6,3,8,9,2,7,7,6,0,2,4,8,2,7,2,2,9,3,5,5,9,4,4,2,3,0,8,8,3,6,8,5,0,0,3,4,9,9,8,9,9,3,4,5,5,9,1,4,1,8,1,8,8,4,0,0,8
%N Decimal expansion of a constant q such that Sum_{n>0} q^(n^2) / (1 + q^n)^(n+1) = 1.
%C Compare to the identity: Sum_{n>=0} t^n/(1 + t)^(n+1) = 1 for all real t > -1.
%C Related series identity: Sum_{n>=0} x^(n^2)/(1 + x^n)^(n+1) = Sum_{n>=0} (x^n - 1)^n, which holds for |x| < 1 and at x = 1.
%C Note that Sum_{n>=0} q^(n^2)/(1 + q^n)^n diverges when q equals this constant.
%C Related constants: a relative maximum for F(x) = Sum_{n>0} x^(n^2) / (1 + x^n)^(n+1) occurs at x = r = 1.16770163525453860038060210814815171759269740752204 61096022701834019548200984085800877983418367920675... where F(r) = 1.62296829171282092185394583034435963782567708182473 69241563842957219935907486317481375662246384816002...; the constant r satisfies Sum_{n>=0} n * (n - r^n) * r^(n^2) / (1 + r^n)^(n+2) = 0.
%F Constant q satisfies:
%F (1) Sum_{n>0} q^(n^2) / (1 + q^n)^(n+1) = 1.
%F (2) Sum_{n>0} q^(-n) / (1 + q^(-n))^(n+1) = 1.
%e The initial 1050 digits of the constant are:
%e q = 2.08512411763439372380336860597510492646644984917005\
%e 60399166820475685459472683380608633685728475391666\
%e 23202960523783396858792345620523112117293556389277\
%e 60248272293559442308836850034998993455914181884008\
%e 17413830379380420723394493519228868838277264250552\
%e 70338374888180842285509880667363656335623958582189\
%e 14957227277741457974426468080521597137811124272934\
%e 77644094270592199652753161086962841342379558889650\
%e 66813332146747026294593263775521540009547253097527\
%e 21223780458855792702371920654676025439770399813608\
%e 58163997909646639377553074980011935193988180130706\
%e 87431850604890853256977074795669925397675297237888\
%e 48538031116570208321040148368549607516080806946967\
%e 19390696127990123894175048822839082258147654679789\
%e 68673370868246837943169347184978182144767139980003\
%e 04843398161679491979027572749436392635882596355424\
%e 88655297144993770936404696899918268972299812682654\
%e 09750091784431323103697192747125489365588143112222\
%e 06559003610924134478070966807827169484545374171016\
%e 15811105252817860965040577295069618649899630322302\
%e 86215867892980222282818894596943887764450079690287....
%e RELATED VALUES.
%e 1/q = 0.4795877576508566835272787486017081382964967858692...
%e where Sum_{n>0} (1/q)^n / (1 + (1/q)^n)^(n+1) = 1.
%e Series Sum_{n>=0} q^(n^2)/(1 + q^n)^n diverges,
%e but: Sum_{n>=0} ( q^(n^2)/(1 + q^n)^n - 1 ) = -1.39414148047935302261469263168...
%K nonn,cons
%O 1,1
%A _Paul D. Hanna_, Nov 21 2018
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