OFFSET
1,1
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1001
FORMULA
EXAMPLE
c = 2.55500248436101360804704969796239525102504151483916927730...
where the constant is equal to the sum
c = 1 + binomial(1,1) + binomial(3/2,2) + binomial(9/4,3) + binomial(25/8,4) + binomial(65/16,5) + binomial(161/32,6) +...+ binomial(n-1 + 1/2^(n-1), n) +...
which may be written as
c = 1 + 2/2 + 6/2^4 + 60/2^9 + 2550/2^16 + 476476/2^25 + 384115732/2^36 + 1305385229720/2^49 + 18382187112952806/2^64 +...+ A224883(n)*x^n/2^(n^2) +...
The constant also equals the logarithmic sum
c = 1 + 2*log(2) + 4*log(4/3)^2/2! + 8*log(8/7)^3/3! + 16*log(16/15)^4/4! + 32*log(32/31)^5/5! + 64*log(64/63)^6/6! +...+ (-2)^n*log(1 - 1/2^n)^n/n! +...
which converges rather quickly.
PROG
(PARI) /* By definition: */
\p128
{c=suminf(n=0, binomial(n-1 + 1/2^(n-1), n)*1.)}
{a(n)=floor(10^n*c)%10}
for(n=0, 120, print1(a(n), ", "))
(PARI) /* By a logarithmic identity (accelerated series): */
\p1024
{c=1+suminf(n=1, (-2)^n*log(1 - 1/2^n)^n / n!)}
{a(n)=floor(10^n*c)%10}
for(n=0, 1000, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paul D. Hanna, Nov 29 2014
STATUS
approved