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A265291 Decimal expansion of Sum_{n >= 1} (x - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

%I A265291 #13 Aug 23 2022 09:52:42
%S A265291 4,2,8,8,6,0,3,3,8,0,6,8,0,9,5,9,8,3,0,0,2,1,1,1,3,6,7,6,1,3,2,7,2,3,
%T A265291 0,7,2,3,9,6,0,1,7,6,5,1,2,5,6,0,8,2,7,4,6,6,8,3,0,2,9,6,0,2,2,3,0,5,
%U A265291 6,9,3,1,3,7,0,6,6,5,3,5,8,8,2,6,1,4
%N A265291 Decimal expansion of Sum_{n >= 1} (x - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).
%F A265291 From _Peter Bala_, Aug 20 2022: (Start)
%F A265291 Equals Sum_{n >= 1} (-1)^(n+1)/Pell(2*n), where Pell(n) = A000129(n).
%F A265291 Equals 2*sqrt(2)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = 2^sqrt(2) - 3. (End)
%e A265291 sum = 0.4288603380680959830021113676132723...
%p A265291 x := 2*sqrt(2) - 3:
%p A265291 evalf(2*sqrt(2)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..16), 100); # _Peter Bala_, Aug 21 2022
%t A265291 x = Sqrt[2]; z = 600; c = Convergents[x, z];
%t A265291 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265291 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265291 N[s1 + s2, 200]
%t A265291 RealDigits[s1, 10, 120][[1]]  (* A265291 *)
%t A265291 RealDigits[s2, 10, 120][[1]]  (* A265292 *)
%t A265291 RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)
%Y A265291 Cf. A000129, A001227, A002193, A265292, A265293, A265288 (guide).
%K A265291 nonn,cons
%O A265291 0,1
%A A265291 _Clark Kimberling_, Dec 06 2015