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Beatty sequence for 2^x, where 1/2^x + 1/3^(x-1) = 1.
3

%I #4 Jan 04 2020 12:58:15

%S 2,5,8,10,13,16,18,21,24,26,29,32,34,37,40,42,45,48,50,53,56,59,61,64,

%T 67,69,72,75,77,80,83,85,88,91,93,96,99,101,104,107,110,112,115,118,

%U 120,123,126,128,131,134,136,139,142,144,147,150,152,155,158,161

%N Beatty sequence for 2^x, where 1/2^x + 1/3^(x-1) = 1.

%C Let x be the solution of 1/2^x + 1/3^(x-1) = 1. Then (floor(n 2^x) and (floor(n 3^(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 2^x), where x = 1.4243198392... is the constant in A330093.

%t r = x /. FindRoot[1/2^x + 1/3^(x - 1) == 1, {x, 1, 10}, WorkingPrecision -> 200]

%t RealDigits[r][[1]] (* A330093 *)

%t Table[Floor[n*2^r], {n, 1, 250}] (* A330094 *)

%t Table[Floor[n*3^(r - 1)], {n, 1, 250}] (* A330095 *)

%Y Cf. A329825, A330093, A330095 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 04 2020