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

%I A228041 #15 May 22 2021 04:28:53
%S A228041 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,
%T A228041 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,
%U A228041 5,1,5,9,4,9,8,3,4,1,6,9,1,8,0,7,7,0
%N A228041 Decimal expansion of sum of reciprocals, row 3 of Wythoff array, W = A035513.
%C A228041 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 A228041 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 A228041 Equals A079586/2 - 5/4. - _Amiram Eldar_, May 22 2021
%e A228041 1/6 + 1/10 + 1/16 + ...  = 0.4299428331215887765860056514594635898444...
%t A228041 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 A228041 n = 3; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
%t A228041 r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
%t A228041 RealDigits[r, 10]
%Y A228041 Cf. A035513, A079586, A228040, A228042, A228043.
%K A228041 nonn,cons,easy
%O A228041 0,1
%A A228041 _Clark Kimberling_, Aug 05 2013