A293380 Constant r satisfies: 0 = Sum_{n>=1} (1/2 - r^n)^n/n.

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, 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, 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

Offset

1,1

Comments

Motivated by the identity: Sum_{n=-oo..+oo, n<>0} (x - y^n)^n/n = -log(1-x), where 0 < |y| < 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Formula

Constant r satisfies:

(1) 0 = Sum_{n>=1} (1/2 - r^n)^n/n.

(2) log(2) = Sum_{n>=1} -(-2)^n * r^(n^2) / (n * (2 - r^n)^n).

(3) log(2) = Sum_{n=-oo..+oo, n<>0} (1/2 - r^n)^n/n.

Example

This constant r satisfies:
(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 +...
(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) +...
Generate this constant by starting with r = 1/2, then iterating:
r = 1/2 + Sum_{n>=2} (1/2 - r^n)^n/n
until desired precision is obtained.
The decimal expansion of this constant begins:
r = 0.54718024396616249220572950633367612507408217123040\
14546897984834967443737682114272275274464764362998\
75833202906534518427398149200315611467742264534933\
07410900052376653487880601550435952256524412957681\
82693465860618497191799083347673481372585407644099\
24055191128326813665663792044619018918015138612919\
22517558095362487924139590714375812254869132031832\
18367379983243100982933520788500322157294335929007\
55655664462513200033351752386548227393277008165715\
29410668980294972340791666277226143340137889105699\
35060868564903372212515078409032998013830380846461\
16660724937698814144627042744975548967453269729505\
00837350332540478154056153357459272811285243101502\
99873154285994445948954150068646715198122601416180\
27269065095980272424381878673803675794878861979766\
52053648913218593538722216325284646073380549624908\
40947592959138732827303377668432579538738949156079\
09323721120215443092493318959352211206766875039409\
58294662290861736158641953284177195304501155824207\
36558392796387833385010708345397097472030780714382...
The binary representation of this constant begins:
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, ...].
The reciprocal of this constant is approximately
1/r = 1.82755136178096331900589049975995341534409500025148884404865962127...

Prog

(PARI) /* Print N digits of constant r (up to precision) */
N=100
{r=.5; for(i=1, 2*N, r = (r + 1/2 + suminf(n=2, (1/2 - r^n)^n/n ))/2); r}
{for(n=1, N, print1( floor(r*10^n)%10, ", "))}  \\ print N digits
{suminf(n=1, -(-2)^n * r^(n^2)/(n * (2 - r^n)^n))}  \\ ~ log(2)

Crossrefs

Sequence in context: A246724 A199276 A373009 * A358663 A021870 A210974

Adjacent sequences: A293377 A293378 A293379 * A293381 A293382 A293383

Keyword

nonn,cons

Author

Paul D. Hanna, Oct 07 2017

Status

approved