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Search: id:A100671
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| A100671 |
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A Graham-Pollak-like sequence with multiplier 3 instead of 2. |
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+0
1
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| 1, 2, 4, 7, 12, 21, 37, 64, 111, 193, 335, 581, 1007, 1745, 3023, 5236, 9069, 15708, 27207, 47124, 81622, 141374, 244867, 424122, 734601, 1272367, 2203805, 3817103, 6611417, 11451311, 19834253, 34353934, 59502760, 103061804, 178508282
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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When the multiplier in the recurrence is 2, we have the Graham-Pollak sequence, where there is a remarkable exact explicit formula for a(n) in terms of the union of the set of integers and the set of integer multiples of Sqrt(2). As Weisstein summarizes Borwein & Bailey: "It is not known if sequences such as a(n) = Floor(Sqrt(3*a(n-1)*(a(n-1)+1))) have corresponding properties." This sequence is the given one, with a(0) = 1. Through n=40, the primes are when n = 1, 3, 6, 9, 14, 25, 28, 29. Through n=40, the semiprimes are when n = 2, 5, 8, 10, 11, 12, 13, 16, 21, 22, 26, 27, 30, 31, 38.
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REFERENCES
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Borwein, J. and Bailey, D., Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. Zbl 201.04705.
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LINKS
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Eric Weisstein's World of Mathematics, Graham-Pollak sequence
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FORMULA
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a(0) = 1, a(n) = Floor(Sqrt(3*a(n-1)*(a(n-1)+1))).
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EXAMPLE
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a(9) = 193 because a(8) = 111; so a(9) = Floor(Sqrt(3*111*(111+1))) = floor(sqrt(37296)) = 193, which happens to be prime.
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CROSSREFS
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Cf. A001521, A091522, A091523.
Sequence in context: A100482 A003293 A094974 this_sequence A005251 A014167 A103197
Adjacent sequences: A100668 A100669 A100670 this_sequence A100672 A100673 A100674
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KEYWORD
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nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 06 2004
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