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Decimal expansion of Sum_{k >= 1} (1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1)), negated.
1

%I #47 Apr 12 2021 16:28:50

%S 0,0,1,0,1,0,1,0,2,0,1,0,1,0,3,0,1,0,1,0,3,0,1,0,2,0,3,0,1,0,1,0,3,0,

%T 3,0,1,0,3,0,1,0,1,0,5,0,1,0,2,0,3,0,1,0,3,0,3,0,1,0,1,0,5,0,3,0,1,0,

%U 3,0,1,0,1,0,5,0,3,0,1,0,4,0,1,0,3,0,3,0,1,0,3,0,3,0,3,0,1,0,5,0

%N Decimal expansion of Sum_{k >= 1} (1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1)), negated.

%C It equals Sum_{k >= 1} 1/((2^(4*k + 2)*5^(4*k + 2)) - 1) - 1/((2^(2*k + 1)*5^(2*k + 1)) - 1).

%C Note that Sum_{k >= 1} (1/(10^k - 1)) / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) -(1/(10^(2*k + 1) - 1))) = A073668 / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1))) = -121.100.

%C My idea for this decimal expansion came from the Engel expansion of e - 1, i.e., A000027(n) = n, and the Engel expansion of e^(-1), i.e., A059193(n) = 2*(2*n + 1)*(n - 1), which I have transformed into (2*n + 1)^2 - (6*n + 3) (since 2*(2*n + 1)*(n - 1) = (2*n + 1)^2 - (6*n + 3)). It appears that the Engel expansion of 1/e works like a Sundaram sieve.

%C Decimal expansion of Sum_{n>=0} (d(2*n+1) - 1)/(10^(2*n+1) - 1), where d = A000005. - _Jianing Song_, Apr 12 2021

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel expansion</a>.

%H English Wikipedia, <a href="https://en.wikipedia.org/wiki/Sieve_of_Sundaram">Sieve of Sundaram</a>.

%H French Wikipedia, <a href="http://fr.wikipedia.org/wiki/Crible_de_Sundaram">Crible de Sundaram</a>.

%e -0.00101010201010301010301020301010303010301010501020301030301...

%o (PARI) suminf(k=1, 1/(10^(4*k + 2) - 1) - 1/(10^(2*k + 1) - 1)) \\ _Michel Marcus_, Jun 25 2019

%K cons,nonn

%O 0,9

%A _Eric Desbiaux_, Apr 06 2009

%E Comments edited by _Petros Hadjicostas_, Jun 19 2019