A100671 A Graham-Pollak-like sequence with multiplier 3 instead of 2.

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

Offset: 0

Author

Jonathan Vos Post, Dec 06 2004

Keywords

• nonn

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.

Examples

			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.
		

References

• Borwein, J. and Bailey, D., Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.

Links

• 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.
• Eric Weisstein's World of Mathematics, Graham-Pollak sequence

Crossrefs

Cf. A001521, A091522, A091523.

Programs

Mathematica

RecurrenceTable[{a[0]==1,a[n]==Floor[Sqrt[3a[n-1](a[n-1]+1)]]},a[n],{n,40}] (* Harvey P. Dale, Sep 10 2011 *)

Formula

a(0) = 1, a(n) = Floor(Sqrt(3*a(n-1)*(a(n-1)+1))).

Extensions

Corrected and extended by Harvey P. Dale, Sep 10 2011