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A255977
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The number of numbers j+k*r <= n, where r = golden ratio and j and k are nonnegative integers.
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3
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1, 2, 4, 6, 9, 13, 17, 22, 27, 33, 40, 47, 55, 64, 73, 83, 93, 104, 116, 128, 141, 154, 168, 183, 198, 214, 231, 248, 266, 284, 303, 323, 343, 364, 386, 408, 431, 454, 478, 503, 528, 554, 580, 607, 635, 663, 692, 722, 752, 783, 814, 846, 879, 912, 946, 980
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OFFSET
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1,2
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COMMENTS
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The difference sequence is A019446.
From Thomas Anton, Oct 22 2018: (Start)
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
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
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.
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
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.
(2), 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
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.
(2), (4), 6, 8, 9, 10, 11, 12, 13, 14, ...
etc.
(End)
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 1..1000
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FORMULA
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a(n) = n + 1 + Sum{floor[(n - k)/tau], k = 0..n}, where tau = (1 + sqrt(5))/2.
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MAPLE
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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
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MATHEMATICA
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Table[n + 1 + Sum[Floor[(n - k)/GoldenRatio], {k, 0, n}], {n, 0, 200}] (*A255977*)
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CROSSREFS
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Cf. A022776, A022778, A022780.
Sequence in context: A025705 A022792 A025697 * A022331 A087483 A154255
Adjacent sequences: A255974 A255975 A255976 * A255978 A255979 A255980
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Mar 14 2015
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STATUS
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approved
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