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A001956 Beatty sequence of (5+sqrt(13))/2.
(Formerly M3327 N1338)
3

%I M3327 N1338 #28 Aug 02 2015 22:18:24

%S 4,8,12,17,21,25,30,34,38,43,47,51,55,60,64,68,73,77,81,86,90,94,98,

%T 103,107,111,116,120,124,129,133,137,141,146,150,154,159,163,167,172,

%U 176,180,185,189,193,197,202,206,210,215,219,223,228,232,236,240,245,249

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

%C 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

%C 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A001956/b001956.txt">Table of n, a(n) for n = 1..10000</a>

%H Ian G. Connell, <a href="http://dx.doi.org/10.4153/CMB-1959-024-3">A generalization of Wythoff's game</a>, Canad. Math. Bull. 2 (1959) 181-190

%H A. S. Fraenkel, <a href="http://www.jstor.org/stable/2321643">How to beat your Wythoff games' opponent on three fronts</a>, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=3)

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

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

%p A001956 := proc(n) local x ; x := (5+sqrt(13))/2 ; floor(n*x) ; end proc:

%p A184480 := proc(n) local x ; x := (sqrt(13)-1)/2 ; floor(n*x) ; end proc:

%p seq(A001956(n),n=1..100) ; # _R. J. Mathar_, Feb 12 2011

%t Table[Floor[n*(5 + Sqrt[13])/2], {n, 100}] (* _T. D. Noe_, Aug 17 2012 *)

%Y Complement of A184480.

%Y Cf. A184117, A184482.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)