A001956 Beatty sequence of (5+sqrt(13))/2.

4, 8, 12, 17, 21, 25, 30, 34, 38, 43, 47, 51, 55, 60, 64, 68, 73, 77, 81, 86, 90, 94, 98, 103, 107, 111, 116, 120, 124, 129, 133, 137, 141, 146, 150, 154, 159, 163, 167, 172, 176, 180, 185, 189, 193, 197, 202, 206, 210, 215, 219, 223, 228, 232, 236, 240, 245, 249

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Inserting a=3 into the Fraenkel formula, a scale factor alpha = (2-a+sqrt(a^2+4))/2 = (sqrt(13)-1)/2 is obtained, which defines the Beatty sequence A184480. The complementary beta parameter, 1/beta+1/alpha=1, is beta = (5+sqrt(13))/2 = 3+alpha, and defines this sequence here, which is the complement in the positive integers. - R. J. Mathar, Feb 12 2011

Upper s-Wythoff sequence, where s(n)=3n. See A184117 for the definition of lower and upper s-Wythoff sequences. - Clark Kimberling, Jan 15 2011

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190
• A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=3)
• Index entries for sequences related to Beatty sequences

Crossrefs

Complement of A184480.

Cf. A184117, A184482.

Programs

Maple

A001956 := proc(n) local x ; x := (5+sqrt(13))/2 ; floor(n*x) ; end proc:
A184480 := proc(n) local x ; x := (sqrt(13)-1)/2 ; floor(n*x) ; end proc:
seq(A001956(n),n=1..100) ; # R. J. Mathar, Feb 12 2011

Mathematica

Table[Floor[n*(5 + Sqrt[13])/2], {n, 100}] (* T. D. Noe, Aug 17 2012 *)

Formula

a(n) = floor(n*beta) with beta = (5+sqrt(13))/2 = 3+(sqrt(13)-1)/2 = 4.30277563773199...