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%I #8 Jun 01 2022 18:10:57
%S 1,3,5,7,8,10,12,14,15,17,19,21,22,24,26,28,29,31,33,35,36,38,40,42,
%T 43,45,47,49,50,52,54,56,57,59,61,63,64,66,68,70,71,73,75,77,78,80,82,
%U 84,85,87,89,91,93,94,96,98,100,101,103,105,107,108,110,112
%N Beatty sequence for 1+x, where 1/(1+x) + 1/(1+x+x^2) = 1.
%C Let x be the positive solution of 1/(1+x) + 1/(1+x+x^2) = 1. Then (floor(n*(1+x)) and (floor(n*(1+x+x^2))) 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))), where x = 0.7548776662... is the constant in A075778.
%t r = x /. FindRoot[1/(1 + x) + 1/(1 + x + x^2) == 1, {x, 1, 2}, WorkingPrecision -> 200]
%t RealDigits[r] (* A075778 *)
%t Table[Floor[n*(1 + r)], {n, 1, 250}] (* A330117 *)
%t Table[Floor[n*(1 + r + r^2)], {n, 1, 250}] (* A330118 *)
%t Plot[1/(1 + x) + 1/(1 + x + x^2) - 1, {x, 0, 2}]
%Y Cf. A075778, A329825, A330118 (complement).
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 04 2020
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