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A252229 The number of numbers j*r^k in the interval [n,n+1), where r = (1 + sqrt(5))/2, the golden ratio, and j >=0, k >= 0.

%I A252229 #5 Dec 18 2014 23:09:22
%S A252229 1,2,2,2,3,2,3,2,3,2,2,3,3,3,2,2,3,3,2,2,3,3,3,2,2,3,2,3,2,4,2,2,2,4,
%T A252229 3,3,2,2,3,2,2,3,3,2,3,2,4,3,2,2,3,2,2,3,3,4,2,2,3,3,2,3,2,3,2,2,3,3,
%U A252229 3,2,2,3,3,2,2,3,4,3,2,2,3,2,3,2,3,2
%N A252229 The number of numbers j*r^k in the interval [n,n+1), where r = (1 + sqrt(5))/2, the golden ratio, and j >=0, k >= 0.
%C A252229 The least n for which a(n) = 4 is 29; the least n for which a(n) = 5 is 199.
%H A252229 Clark Kimberling, <a href="/A252229/b252229.txt">Table of n, a(n) for n = 0..1000</a>
%F A252229 a(n) = 1 + sum{s(n+1,j) - s(n,j), j=1..floor[(n+1)/r]}, where s(n,j) = floor[log(n/j)/log(r)], for n >= 1.
%e A252229 in [0,1):  0
%e A252229 in [1,2):  1, 1 + r
%e A252229 in [2,3):  2, 2 + r
%e A252229 in [3,4):  3, 1+2*r
%e A252229 in [4,5):  4, 1+3*r, 2 + r
%t A252229 z = 100; r = (1 + Sqrt[5])/2;
%t A252229 s[n_, j_] := s[n, j] = Floor[Log[n/j]/Log[r]];
%t A252229 a[n_] := a[n] = Sum[s[n + 1, j] - s[n, j], {j, 1, Floor[(n + 1)/r]}];
%t A252229 t = Join[{1}, Table[1 + a[n], {n, 1, z}]] (* A252229 *)
%Y A252229 Cf. A182801, A020959.
%K A252229 nonn,easy
%O A252229 0,2
%A A252229 _Clark Kimberling_, Dec 16 2014