A329975 - OEIS

%I #12 Dec 18 2024 22:55:40

%S 4,8,12,16,20,24,28,32,36,40,44,48,53,57,61,65,69,73,77,81,85,89,93,

%T 97,101,106,110,114,118,122,126,130,134,138,142,146,150,155,159,163,

%U 167,171,175,179,183,187,191,195,199,203,208,212,216,220,224,228,232

%N Beatty sequence for 1 + x + x^2, where x is the real solution of 1/x + 1/(1+x+x^2) = 1.

%C Let x be the real solution of 1/x + 1/(1+x+x^2) = 1. Then (floor(n*x)) and (floor(n*(x^2 + x + 1))) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>

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

%F a(n) = floor(n*(1+x+x^2)), where x = 1.324717... is the constant in A060006.

%t Solve[1/x + 1/(1 + x + x^2) == 1, x]

%t u = 1/3 (27/2 - (3 Sqrt[69])/2)^(1/3) + (1/2 (9 + Sqrt[69]))^(1/3)/3^(2/3);

%t u1 = N[u, 150]

%t RealDigits[u1, 10][[1]]  (* A060006 *)

%t Table[Floor[n*u], {n, 1, 50}]              (* A329974 *)

%t Table[Floor[n*(1 + u + u^2)], {n, 1, 50}]  (* A329975 *)

%t Plot[1/x + 1/(1 + x + x^2) - 1, {x, -2, 2}]

%Y Cf. A329825, A060006, A329974 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 02 2020