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A096614 Decimal expansion of Sum_{n>=1} f(2^n)/2^n, where f(n) is the number of even digits in n.

%I A096614 #31 Feb 16 2025 08:32:53
%S A096614 1,0,3,1,6,0,6,3,8,6,4,4,5,0,9,6,1,2,2,5,1,5,4,7,7,3,5,4,1,8,7,1,3,0,
%T A096614 3,1,0,3,9,0,2,2,6,4,1,5,2,9,2,6,9,4,0,7,0,9,5,7,6,7,3,2,4,1,2,1,1,1,
%U A096614 0,7,2,8,3,9,2,1,4,0,7,8,9,1,6,0,5,5,6,1,7,2,3,7,5,1,1,2,0,6,8,2,4,0,0,2,5,5
%N A096614 Decimal expansion of Sum_{n>=1} f(2^n)/2^n, where f(n) is the number of even digits in n.
%C A096614 This constant is transcendental. If the number of even digits is replaced with the number of odd digits, then the sum will be 1/9. (Borwein et al. 2004). - _Amiram Eldar_, Nov 14 2020
%D A096614 Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, 2004, pp. 14-15.
%H A096614 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitCount.html">Digit Count</a>.
%H A096614 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A096614 Equals -1/9 + Sum_{k>=1} (1 + floor(k * log_10(2)))/2^k. - _Amiram Eldar_, Nov 14 2020
%e A096614 1.03160638...
%t A096614 RealDigits[-1/9 + Sum[(1 + Floor[k*Log10[2]])/2^k, {k, 1, 350}], 10,
%t A096614 100][[1]] (* _Amiram Eldar_, Nov 14 2020 *)
%o A096614 (PARI) -1/9 + suminf(k=1, (1 + floor(k * log(2)/log(10)))/2^k) \\ _Michel Marcus_, Nov 14 2020
%Y A096614 Cf. A055253.
%K A096614 nonn,cons,base
%O A096614 1,3
%A A096614 _Eric W. Weisstein_, Jun 30 2004