A121474 A devil's staircase constant: decimal expansion of the sums involving powers of 2 and Beatty sequences given by: c = Sum_{n>=1} [log_2(e^n)]/2^n = Sum_{n>=1} 1/2^[log(2^n)].

2, 3, 3, 0, 7, 2, 4, 0, 7, 0, 4, 5, 0, 0, 9, 7, 8, 4, 7, 3, 5, 7, 2, 7, 2, 6, 4, 0, 1, 7, 8, 0, 9, 3, 5, 3, 8, 6, 0, 3, 1, 4, 8, 6, 1, 0, 1, 4, 3, 8, 7, 5, 6, 5, 0, 3, 2, 1, 0, 8, 2, 4, 3, 3, 1, 6, 6, 7, 2, 1, 0, 5, 5, 0, 5, 8, 6, 4, 0, 0, 5, 0, 3, 8, 2, 0, 0, 0, 6, 2, 3, 0, 8, 5, 2, 3, 5, 4, 2, 4, 8, 9, 2, 8, 1

Offset

1,1

Comments

The continued fraction (A121475) of this constant has large partial quotients: c = [2; 3, 42, 4, 4512412933881984, ...]. See the MathWorld link for more information regarding devil's staircase constants. The dual constant is: A121472 = Sum_{n>=1} 1/2^[log_2(e^n)] = Sum_{n>=1} [log(2^n)]/2^n.

Links

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Devil's Staircase

Formula

c = Sum_{n>=1} [n/log(2)]/2^n = Sum_{n>=1} 1/2^[n*log(2)], where [z]=floor(z).

Example

c=2.3307240704500978473572726401780935386031486101438756503210824331667...

Prog

(PARI) a(n)=local(t=log(2), x=sum(m=1, 10*(n+1), floor(m/t)/2^m)); floor(10^n*x)%10
(PARI) a(n)=local(t=log(2), x=sum(m=1, 10*(n+1), 1/2^floor(m*t))); floor(10^n*x)%10

Crossrefs

Cf. A121475 (continued fraction), A121472 (dual constant), A121473.

Sequence in context: A091408 A193382 A106242 * A138003 A329232 A057682

Adjacent sequences: A121471 A121472 A121473 * A121475 A121476 A121477

Keyword

cons,nonn

Author

Paul D. Hanna, Aug 01 2006

Status

approved