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%I #6 Jan 10 2024 16:13:41
%S 2,8,9,9,3,2,1,8,5,6,6,4,4,7,9,5,7,4,0,7,4,1,4,7,5,1,2,3,4,6,1,5,1,0,
%T 5,8,8,1,3,1,7,0,9,3,9,4,5,2,9,1,2,1,6,1,9,8,7,9,1,7,8,5,1,4,4,8,7,2,
%U 5,7,6,1,5,0,1,7,9,7,2,2,0,0,0,1,9,6
%N Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n+1)*g), where g = 1.790044015672757..., s(n) = (s(n - 1) + sqrt(2))^(1/2), s(0) = 2.
%C (lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - sqrt(2). See A298512 for a guide to related sequences.
%e s(0) + s(1) + ... + s(n) - (n+1)*g -> 0.289932185664479574074147512346151058813...
%t s[0] = 2; d = Sqrt[2]; p = 1/2;
%t g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
%t s[n_] := s[n] = (s[n - 1] + d)^p
%t N[Table[s[n], {n, 0, 30}]]
%t s = N[Sum[- g + s[n], {n, 0, 200}], 150 ];
%t RealDigits[s, 10][[1]] (* A298525 *)
%Y Cf. A298512, A298524.
%K nonn,easy,cons
%O 0,1
%A _Clark Kimberling_, Feb 12 2018
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