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%I #11 May 22 2021 04:33:38
%S 2,8,6,6,2,8,5,5,5,4,1,4,3,9,2,5,1,7,7,2,4,0,0,3,7,6,7,6,3,9,6,4,2,3,
%T 9,3,2,2,9,6,3,5,0,4,4,5,7,7,3,2,2,8,2,8,8,3,1,8,5,1,2,7,1,7,7,5,0,4,
%U 3,4,3,9,6,6,5,5,6,1,1,2,7,8,7,1,8,0
%N Decimal expansion of sum of reciprocals, row 4 of Wythoff array, W = A035513.
%C 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 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 Equals A079586/3 - 5/6. - _Amiram Eldar_, May 22 2021
%e 1/9 + 1/15 + 1/24 + ... = 0.28662855541439251772400376763964239322963...
%t 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 n = 4; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
%t r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
%t RealDigits[r, 10]
%Y Cf. A035513, A079586, A228040, A228041, A228043.
%K nonn,cons,easy
%O 0,1
%A _Clark Kimberling_, Aug 05 2013
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