A298523 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n + 1)*g), where g = 1.86676039917386..., s(n) = (s(n - 1) + (1+sqrt(5))/2)^(1/2), s(0) = 2.

1, 8, 1, 4, 9, 0, 0, 8, 3, 3, 3, 4, 2, 5, 0, 7, 8, 0, 8, 2, 2, 5, 3, 9, 3, 1, 2, 6, 3, 7, 4, 2, 1, 9, 8, 4, 3, 5, 7, 7, 0, 3, 6, 3, 4, 3, 7, 5, 9, 7, 0, 3, 7, 2, 4, 9, 9, 5, 1, 2, 3, 0, 6, 2, 4, 0, 8, 4, 1, 8, 3, 7, 8, 5, 4, 4, 3, 7, 2, 2, 5, 5, 6, 6, 6, 2

Offset: 0

Author

Clark Kimberling, Feb 12 2018

Keywords

• nonn
• easy
• cons

Comments

(lim_ {n->oo} s(n)) = g = real zero of x^2 - x - (1+sqrt(5))/2. See A298512 for a guide to related sequences.

Examples

			s(0) + s(1) + ... + s(n) - (n + 1)*g -> 0.1814900833342507808225393126374219...
		

Crossrefs

Cf. A298512, A298522.

Programs

Mathematica

s[0] = 2; d = GoldenRatio; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[- g + s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298523 *)