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Decimal expansion of Sum_{k >= 0} (10^5*A196563(k) - A196564(k)/10^5)/10^k.
1

%I #15 May 20 2024 02:08:59

%S 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 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 0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9,0,9

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

%C 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 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 Paolo Xausa, <a href="/A372947/b372947.txt">Table of n, a(n) for n = 6..10000</a>

%H 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 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 Approximately 11111111111/110000, correct to 109 digits: see Entry 5 in Bradshaw and Vignat (2023), pp. 1 and 10-12.

%e 101010.10100909090909090909090909090909090909090909090909090...

%t First[RealDigits[Sum[(10^5*Count[IntegerDigits[k], _?EvenQ] - Count[IntegerDigits[k], _?OddQ]/10^5)/10^k, {k, 0, 100}], 10, 100]]

%Y Cf. A196563, A196564, A372551.

%K nonn,cons

%O 6,12

%A _Paolo Xausa_, May 17 2024