A121475 - OEIS

%I #6 Feb 16 2025 08:33:02

%S 2,3,42,4,4512412933881984,2722258935367507708887588480171556995584,

%T 2305843009213693952,

%U 6277101735386680763835789423207666416102355444464034512896

%N Partial quotients of the continued fraction expansion of the constant A121474 defined by the sums: c = Sum_{n>=1} [log_2(e^n)]/2^n = Sum_{n>=1} 1/2^[log(2^n)].

%C A "devil's staircase" type of constant has large partial quotients in its continued fraction expansion. See MathWorld link for more information.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DevilsStaircase.html">Devil's Staircase</a>

%e c=2.330724070450097847357272640178093538603148610143875650321...

%e The number of 1's in the binary expansion of a(n) is given by

%e the partial quotients of continued fraction of 1/log(2):

%e 1/log(2) = [1; 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, 1, ...]

%e as can be seen by the binary expansion of a(n):

%e a(0) = 2^1

%e a(1) = 2^1 + 2^0

%e a(2) = 2^5 + 2^3 + 2^1

%e a(3) = 2^2

%e a(4) = 2^52 + 2^43 + 2^34 + 2^25 + 2^16 + 2^7

%e a(5) = 2^131 + 2^70 + 2^9

%e a(6) = 2^61

%e a(7) = 2^192

%e a(8) = 2^698 + 2^253

%e a(9) = 2^445

%e a(10) = 2^1143

%e a(11) = 2^1588

%e a(12) = 2^2731

%e a(13) = 2^18419 + 2^11369 + 2^4319

%Y Cf. A121474 (decimal expansion), A121472 (dual constant), A121473.

%K cofr,nonn,changed

%O 0,1

%A _Paul D. Hanna_, Aug 01 2006