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A329974 Beatty sequence for the real solution x of 1/x + 1/(1+x+x^2) = 1.

%I A329974 #13 Feb 16 2025 08:33:58
%S A329974 1,2,3,5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,27,29,30,31,33,
%T A329974 34,35,37,38,39,41,42,43,45,46,47,49,50,51,52,54,55,56,58,59,60,62,63,
%U A329974 64,66,67,68,70,71,72,74,75,76,78,79,80,82,83,84,86,87
%N A329974 Beatty sequence for the real solution x of 1/x + 1/(1+x+x^2) = 1.
%C A329974 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 A329974 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>
%H A329974 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A329974 a(n) = floor(n*x), where x = 1.324717... is the constant in A060006.
%t A329974 Solve[1/x + 1/(1 + x + x^2) == 1, x]
%t A329974 u = 1/3 (27/2 - (3 Sqrt[69])/2)^(1/3) + (1/2 (9 + Sqrt[69]))^(1/3)/3^(2/3);
%t A329974 u1 = N[u, 150]
%t A329974 RealDigits[u1, 10][[1]]  (* A060006 *)
%t A329974 Table[Floor[n*u], {n, 1, 50}]              (* A329974 *)
%t A329974 Table[Floor[n*(1 + u + u^2)], {n, 1, 50}]  (* A329975 *)
%t A329974 Plot[1/x + 1/(1 + x + x^2) - 1, {x, -2, 2}]
%Y A329974 Cf. A329825, A060006, A329975 (complement).
%K A329974 nonn,easy
%O A329974 1,2
%A A329974 _Clark Kimberling_, Jan 02 2020