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

%I #4 Jan 05 2020 08:11:37

%S 1,3,4,6,7,9,11,12,14,15,17,19,20,22,23,25,27,28,30,31,33,35,36,38,39,

%T 41,43,44,46,47,49,51,52,54,55,57,58,60,62,63,65,66,68,70,71,73,74,76,

%U 78,79,81,82,84,86,87,89,90,92,94,95,97,98,100,102,103

%N Beatty sequence for 3^(x-1), 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 3^(x-1)), 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, A330094, A330095 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 04 2020