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%I #6 Jan 10 2024 16:06:42
%S 5,4,6,3,7,2,3,3,4,7,7,9,8,8,8,4,5,2,8,6,5,8,4,4,0,5,5,1,8,6,1,6,4,7,
%T 8,8,0,2,8,7,5,4,7,7,9,2,0,6,8,8,0,6,2,4,5,6,6,9,2,0,5,5,4,4,7,9,0,6,
%U 9,2,2,8,3,4,9,9,3,9,6,5,3,4,1,0,6,2
%N Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n + 1)*g), where g = (1 + sqrt (5))/2, s(n) = (s(n - 1) + 1)^(1/2), s(0) = 2.
%C (lim_ {n->oo} s(n)) = g = golden ratio, A001622. See A298512 for a guide to related sequences.
%e s(n) -> g = (1+sqrt(5))/2, as at A001622.
%e s(0) + s(1) + ... + s(n) - (n + 1)*g -> 0.54637233477988845286584405518616478...
%t s[0] = 2; d = 1; p = 1/2; s[n_] := s[n] = (s[n - 1] + d)^p
%t N[Table[s[n], {n, 0, 30}]]
%t z = 200 ; g = GoldenRatio; s = N[-(z + 1)*g + Sum[s[n], {n, 0, z}], 150 ];
%t RealDigits[s, 10][[1]]; (* A298513 *)
%Y Cf. A001622, A298512, A298514.
%K nonn,easy,cons
%O 0,1
%A _Clark Kimberling_, Feb 11 2018
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