|
| |
|
|
A100671
|
|
A Graham-Pollak-like sequence with multiplier 3 instead of 2.
|
|
0
| |
|
|
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, 59502759, 103061802, 178508278, 309185407, 535524834
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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.
|
|
|
REFERENCES
| 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.
|
|
|
LINKS
| Eric Weisstein's World of Mathematics, Graham-Pollak sequence
|
|
|
FORMULA
| a(0) = 1, a(n) = Floor(Sqrt(3*a(n-1)*(a(n-1)+1))).
|
|
|
EXAMPLE
| 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.
|
|
|
MATHEMATICA
| RecurrenceTable[{a[0]==1, a[n]==Floor[Sqrt[3a[n-1](a[n-1]+1)]]}, a[n], {n, 40}] (* From Harvey P. Dale, Sep 10 2011 *)
|
|
|
CROSSREFS
| Cf. A001521, A091522, A091523.
Sequence in context: A192759 A094974 A189593 * A189600 A005251 A014167
Adjacent sequences: A100668 A100669 A100670 * A100672 A100673 A100674
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 06 2004
|
|
|
EXTENSIONS
| Corrected and extended by Harvey P. Dale, Sep 10 2011
|
| |
|
|
