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A228040 Decimal expansion of sum of reciprocals, row 2 of Wythoff array, W = A035513.

%I A228040 #12 May 22 2021 04:29:00
%S A228040 6,2,9,5,2,4,8,3,9,8,7,6,3,1,2,4,4,9,5,3,5,4,6,1,7,9,5,3,4,1,8,5,0,1,
%T A228040 9,3,3,1,6,2,5,9,6,8,3,8,2,8,8,8,6,0,8,7,7,9,7,3,8,1,9,0,7,0,8,3,7,2,
%U A228040 8,2,7,4,2,1,3,1,2,7,0,4,4,6,4,5,7,0
%N A228040 Decimal expansion of sum of reciprocals, row 2 of Wythoff array, W = A035513.
%C A228040 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.
%C A228040 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.
%F A228040 Equals A093540 - 4/3. - _Amiram Eldar_, May 22 2021
%e A228040 1/4 + 1/7 + 1/11 + ... = 0.629524839876312449535461795341...
%t A228040 f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
%t A228040 n = 2; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
%t A228040 r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
%t A228040 RealDigits[r, 10]
%Y A228040 Cf. A035513, A079586, A093540, A228041, A228042, A228043.
%K A228040 nonn,cons,easy
%O A228040 0,1
%A A228040 _Clark Kimberling_, Aug 05 2013