A228041 Decimal expansion of sum of reciprocals, row 3 of Wythoff array, W = A035513.

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

Offset: 0

Author

Clark Kimberling, Aug 05 2013

Keywords

• nonn
• cons
• easy

Comments

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

Examples

			1/6 + 1/10 + 1/16 + ...  = 0.4299428331215887765860056514594635898444...
		

Crossrefs

Cf. A035513, A079586, A228040, A228042, A228043.

Programs

Mathematica

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 3; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Formula

Equals A079586/2 - 5/4. - Amiram Eldar, May 22 2021