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A372947 Decimal expansion of Sum_{k >= 0} (10^5*A196563(k) - A196564(k)/10^5)/10^k.

%I A372947 #15 May 20 2024 02:08:59
%S A372947 1,0,1,0,1,0,1,0,1,0,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,
%T A372947 0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,
%U A372947 0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9
%N A372947 Decimal expansion of Sum_{k >= 0} (10^5*A196563(k) - A196564(k)/10^5)/10^k.
%C A372947 Bradshaw and Vignat (2023, p. 12) show that, in general, for p >= 1 Sum_{k >= 0} c_p(k)/10^(p*k) can be approximated by [1[0]_{p-1}]_{10}1/1[0]_{p-1}1[0]_{4*p} with an error on the order of 10^(-105*p), where c_p(k) = 10^(p*5)*A196563(k) - A196564(k)/10^(p*5) and [x]_{r} denotes r copies of x.
%C A372947 E.g., for p = 2 we have that Sum_{k >= 0} c_2(k)/10^(2*k) = Sum_{k >= 0} (10^(2*5)*A196563(k) - A196564(k)/10^(2*5))/10^(2*k) can be approximated by 101010101010101010101/10100000000.
%H A372947 Paolo Xausa, <a href="/A372947/b372947.txt">Table of n, a(n) for n = 6..10000</a>
%H A372947 J. M. Borwein and P. B. Borwein, <a href="https://doi.org/10.2307/2324993">Strange Series and High Precision Fraud</a>, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.
%H A372947 Zachary P. Bradshaw and Christophe Vignat, <a href="https://doi.org/10.48550/arXiv.2307.05565">Dubious Identities: A Visit to the Borwein Zoo</a>, arXiv:2307.05565v1 [math.HO], 2023.
%F A372947 Approximately 11111111111/110000, correct to 109 digits: see Entry 5 in Bradshaw and Vignat (2023), pp. 1 and 10-12.
%e A372947 101010.10100909090909090909090909090909090909090909090909090...
%t A372947 First[RealDigits[Sum[(10^5*Count[IntegerDigits[k], _?EvenQ] - Count[IntegerDigits[k], _?OddQ]/10^5)/10^k, {k, 0, 100}], 10, 100]]
%Y A372947 Cf. A196563, A196564, A372551.
%K A372947 nonn,cons
%O A372947 6,12
%A A372947 _Paolo Xausa_, May 17 2024