cp's OEIS Frontend

A255977 The number of numbers j+k*r <= n, where r = golden ratio and j and k are nonnegative integers.

%I A255977 #34 Dec 11 2024 13:55:43
%S A255977 1,2,4,6,9,13,17,22,27,33,40,47,55,64,73,83,93,104,116,128,141,154,
%T A255977 168,183,198,214,231,248,266,284,303,323,343,364,386,408,431,454,478,
%U A255977 503,528,554,580,607,635,663,692,722,752,783,814,846,879,912,946,980
%N A255977 The number of numbers j+k*r <= n, where r = golden ratio and j and k are nonnegative integers.
%C A255977 The difference sequence is A019446.
%C A255977 From _Thomas Anton_, Oct 22 2018: (Start)
%C A255977 It appears that this sequence (apart from the first term) can be obtained through the following sieving process. Start with the positive integers. Then, at each stage, circle the first remaining number that has not already been circled, and delete all terms in the subsequence of terms that were not circled in previous stages with circled indices that have not yet been deleted. E.g., the first few iterations are
%C A255977   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
%C A255977 We circle 1, and take the subsequence of previously uncircled numbers, which is the entire sequence, and delete all terms with circled indices that have not been deleted, in this case, just the 1st, 1.
%C A255977   2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
%C A255977 We circle 2, and take the previously uncircled subsequence, which is again the entire sequence, and delete all terms with circled indices that have not been deleted, just the 2nd, 3.
%C A255977   (2), 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
%C A255977 We circle 4, and take the previously uncircled subsequence, which is all terms of the sequence except 2, and delete all terms in that subsequence with circled indices (as terms of the subsequence) that have not been deleted, the 2nd and 4th, respectively the 3rd and 5th terms of the entire sequence, 5 and 7.
%C A255977   (2), (4), 6, 8, 9, 10, 11, 12, 13, 14, ...
%C A255977 etc.
%C A255977 (End)
%H A255977 Clark Kimberling, <a href="/A255977/b255977.txt">Table of n, a(n) for n = 1..1000</a>
%F A255977 a(n) = n + 1 + Sum{floor[(n - k)/tau], k = 0..n}, where tau = (1 + sqrt(5))/2.
%F A255977 a(n) = A054347(n-1) - (n^2 - 3*n)/2. - _Alan Michael Gómez Calderón_, Nov 21 2024
%p A255977 t:=(1+sqrt(5))/2: a:=n->n+1+add(floor((n-k)/t),k=0..n): seq(a(n),n=0..55); # _Muniru A Asiru_, Oct 24 2018
%t A255977 Table[n + 1 + Sum[Floor[(n - k)/GoldenRatio], {k, 0, n}], {n, 0, 200}]
%Y A255977 Cf. A022776, A022778, A022780, A054347.
%K A255977 nonn,easy
%O A255977 1,2
%A A255977 _Clark Kimberling_, Mar 14 2015