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A321599
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Decimal expansion of a constant q such that Sum_{n>0} q^(n^2) / (1 + q^n)^(n+1) = 1.
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0
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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, 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, 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
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OFFSET
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1,1
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COMMENTS
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Compare to the identity: Sum_{n>=0} t^n/(1 + t)^(n+1) = 1 for all real t > -1.
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.
Note that Sum_{n>=0} q^(n^2)/(1 + q^n)^n diverges when q equals this constant.
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.
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LINKS
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FORMULA
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Constant q satisfies:
(1) Sum_{n>0} q^(n^2) / (1 + q^n)^(n+1) = 1.
(2) Sum_{n>0} q^(-n) / (1 + q^(-n))^(n+1) = 1.
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EXAMPLE
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The initial 1050 digits of the constant are:
q = 2.08512411763439372380336860597510492646644984917005\
60399166820475685459472683380608633685728475391666\
23202960523783396858792345620523112117293556389277\
60248272293559442308836850034998993455914181884008\
17413830379380420723394493519228868838277264250552\
70338374888180842285509880667363656335623958582189\
14957227277741457974426468080521597137811124272934\
77644094270592199652753161086962841342379558889650\
66813332146747026294593263775521540009547253097527\
21223780458855792702371920654676025439770399813608\
58163997909646639377553074980011935193988180130706\
87431850604890853256977074795669925397675297237888\
48538031116570208321040148368549607516080806946967\
19390696127990123894175048822839082258147654679789\
68673370868246837943169347184978182144767139980003\
04843398161679491979027572749436392635882596355424\
88655297144993770936404696899918268972299812682654\
09750091784431323103697192747125489365588143112222\
06559003610924134478070966807827169484545374171016\
15811105252817860965040577295069618649899630322302\
86215867892980222282818894596943887764450079690287....
RELATED VALUES.
1/q = 0.4795877576508566835272787486017081382964967858692...
where Sum_{n>0} (1/q)^n / (1 + (1/q)^n)^(n+1) = 1.
Series Sum_{n>=0} q^(n^2)/(1 + q^n)^n diverges,
but: Sum_{n>=0} ( q^(n^2)/(1 + q^n)^n - 1 ) = -1.39414148047935302261469263168...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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