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A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.

%I A228043 #14 May 22 2021 04:28:32
%S A228043 2,1,4,9,7,1,4,1,6,5,6,0,7,9,4,3,8,8,2,9,3,0,0,2,8,2,5,7,2,9,7,3,1,7,
%T A228043 9,4,9,2,2,2,2,6,2,8,3,4,3,2,9,9,2,1,2,1,6,2,3,8,8,8,4,5,3,8,3,1,2,8,
%U A228043 2,5,7,9,7,4,9,1,7,0,8,4,5,9,0,3,8,5
%N A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.
%C A228043 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 A228043 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 A228043 Equals A079586/4 - 5/8. - _Amiram Eldar_, May 22 2021
%e A228043 1/12 + 1/20 + 1/32 + ... = 0.21497141656079438829300282572973179492222...
%t A228043 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 A228043 n = 5; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
%t A228043 r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
%t A228043 RealDigits[r, 10]
%Y A228043 Cf. A035513, A079586, A228040, A228041, A228042.
%K A228043 nonn,cons,easy
%O A228043 0,1
%A A228043 _Clark Kimberling_, Aug 05 2013