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A080754
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a(n) = ceiling(n*(1+sqrt(2))).
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6
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3, 5, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 37, 39, 42, 44, 46, 49, 51, 54, 56, 58, 61, 63, 66, 68, 71, 73, 75, 78, 80, 83, 85, 87, 90, 92, 95, 97, 99, 102, 104, 107, 109, 112, 114, 116, 119, 121, 124, 126, 128, 131, 133, 136, 138, 141, 143, 145
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OFFSET
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1,1
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COMMENTS
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Positive integer solutions to the equation x = ceiling(r*floor(x/r)), where r = 1+sqrt(2). - Benoit Cloitre, Feb 14 2004
Equivalently, numbers m such that {rm} <= {r}, where r=2^(1/2) and { } denotes fractional part.
Andrew Plewe, May 18 2007, observed that the sequence defined by a(n) = ceiling(n*(1+sqrt(2))) appeared to give the same numbers as the sequence, originally due to Clark Kimberling, Jul 01 2006, defined by: numbers m such that {rm} <= {r}, where r=2^(1/2). That these sequences are indeed the same was shown by David Applegate. This follows since the complements of the two sequences are the same, which is shown in the comments on A080755.
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LINKS
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FORMULA
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a(1) = 3; for n>1, a(n) = a(n-1) + 3 if n is in sequence, a(n) = a(n-1) + 2 if not.
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MATHEMATICA
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Table[Ceiling[n*(1 + Sqrt[2])], {n, 1, 50}] (* G. C. Greubel, Nov 28 2017 *)
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PROG
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(PARI) for(n=1, 30, print1(ceil(n*(1+sqrt(2))), ", ")) \\ G. C. Greubel, Nov 28 2017
(Magma) [Ceiling(n*(1+Sqrt(2))): n in [1..30]]; // G. C. Greubel, Nov 28 2017
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CROSSREFS
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Equals A003151 + 1. This and its complement A080755 partition the integers >= 2.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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