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

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, 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, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9

Offset: 6

Author

Paolo Xausa, May 17 2024

Keywords

• nonn
• cons

Comments

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.

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.

Examples

			101010.10100909090909090909090909090909090909090909090909090...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 6..10000
• J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.
• Zachary P. Bradshaw and Christophe Vignat, Dubious Identities: A Visit to the Borwein Zoo, arXiv:2307.05565v1 [math.HO], 2023.

Crossrefs

Cf. A196563, A196564, A372551.

Programs

Mathematica

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

Formula

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