%I #21 Oct 07 2017 23:43:08
%S 5,4,7,1,8,0,2,4,3,9,6,6,1,6,2,4,9,2,2,0,5,7,2,9,5,0,6,3,3,3,6,7,6,1,
%T 2,5,0,7,4,0,8,2,1,7,1,2,3,0,4,0,1,4,5,4,6,8,9,7,9,8,4,8,3,4,9,6,7,4,
%U 4,3,7,3,7,6,8,2,1,1,4,2,7,2,2,7,5,2,7,4,4,6,4,7,6,4,3,6,2,9,9,8,7,5,8,3,3,2,0,2,9,0,6,5,3,4,5,1,8,4,2,7,3,9,8,1,4,9,2,0,0,3,1,5,6,1,1,4,6,7,7,4,2,2,6,4,5,3,4,9,3,3,0,7,4,1,0,9,0,0,0,5,2,3,7,6,6,5,3,4,8,7,8,8,0,6,0,1,5,5,0,4,3,5,9,5,2,2,5,6,5,2,4,4,1,2,9,5,7,6,8,1
%N Constant r satisfies: 0 = Sum_{n>=1} (1/2 - r^n)^n/n.
%C Motivated by the identity: Sum_{n=-oo..+oo, n<>0} (x - y^n)^n/n = -log(1-x), where 0 < |y| < 1.
%H Paul D. Hanna, <a href="/A293380/b293380.txt">Table of n, a(n) for n = 1..1000</a>
%F Constant r satisfies:
%F (1) 0 = Sum_{n>=1} (1/2 - r^n)^n/n.
%F (2) log(2) = Sum_{n>=1} -(-2)^n * r^(n^2) / (n * (2 - r^n)^n).
%F (3) log(2) = Sum_{n=-oo..+oo, n<>0} (1/2 - r^n)^n/n.
%e This constant r satisfies:
%e (1) 0 = (1/2 - r) + (1/2 - r^2)^2/2 + (1/2 - r^3)^3/3 + (1/2 - r^4)^4/4 + (1/2 - r^5)^5/5 + (1/2 - r^6)^6/6 + (1/2 - r^7)^7/7 +...+ (1/2 - r^n)^n/n +...
%e (2) log(2) = 2*r/(1*(2-r)) - 4*r^4/(2*(2-r^2)^2) + 8*r^9/(3*(2-r^3)^3) - 16*r^16/(4*(2-r^4)^4) + 32*r^25/(5*(2-r^5)^5) - 64*r^36/(6*(2-r^6)^6) + 128*r^49/(7*(2-r^7)^7) +...+ -(-2)^n*r^(n^2)/(n*(2 - r^n)^n) +...
%e Generate this constant by starting with r = 1/2, then iterating:
%e r = 1/2 + Sum_{n>=2} (1/2 - r^n)^n/n
%e until desired precision is obtained.
%e The decimal expansion of this constant begins:
%e r = 0.54718024396616249220572950633367612507408217123040\
%e 14546897984834967443737682114272275274464764362998\
%e 75833202906534518427398149200315611467742264534933\
%e 07410900052376653487880601550435952256524412957681\
%e 82693465860618497191799083347673481372585407644099\
%e 24055191128326813665663792044619018918015138612919\
%e 22517558095362487924139590714375812254869132031832\
%e 18367379983243100982933520788500322157294335929007\
%e 55655664462513200033351752386548227393277008165715\
%e 29410668980294972340791666277226143340137889105699\
%e 35060868564903372212515078409032998013830380846461\
%e 16660724937698814144627042744975548967453269729505\
%e 00837350332540478154056153357459272811285243101502\
%e 99873154285994445948954150068646715198122601416180\
%e 27269065095980272424381878673803675794878861979766\
%e 52053648913218593538722216325284646073380549624908\
%e 40947592959138732827303377668432579538738949156079\
%e 09323721120215443092493318959352211206766875039409\
%e 58294662290861736158641953284177195304501155824207\
%e 36558392796387833385010708345397097472030780714382...
%e The binary representation of this constant begins:
%e binary(r) = [1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, ...].
%e The reciprocal of this constant is approximately
%e 1/r = 1.82755136178096331900589049975995341534409500025148884404865962127...
%o (PARI) /* Print N digits of constant r (up to precision) */
%o N=100
%o {r=.5; for(i=1,2*N, r = (r + 1/2 + suminf(n=2, (1/2 - r^n)^n/n ))/2); r}
%o {for(n=1,N,print1( floor(r*10^n)%10,", "))} \\ print N digits
%o {suminf(n=1, -(-2)^n * r^(n^2)/(n * (2 - r^n)^n))} \\ ~ log(2)
%K nonn,cons
%O 1,1
%A _Paul D. Hanna_, Oct 07 2017
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