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A330002 Beatty sequence for x, where 1/x + 1/(x+1)^2 = 1. 3

%I #12 Jan 27 2020 04:30:54

%S 1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24,26,27,28,29,31,

%T 32,33,34,36,37,38,39,41,42,43,44,46,47,48,49,51,52,53,54,56,57,58,59,

%U 61,62,63,64,66,67,68,69,71,72,73,74,76,77,78,79,81,82,83

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

%C Let x be the solution of 1/x + 1/(x+1)^2 = 1. Then (floor(n x) and (floor(n (x+1)^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.

%C Differs from A047201 first at A047201(85)=106, a(85)=105. - _R. J. Mathar_, Jan 11 2020

%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*x), where x = 1.24697960371... is the constant in A255249.

%p A330002 := proc(n)

%p local x;

%p x := -2*cos(5*Pi/7) ;

%p floor(n*x) ;

%p end proc: # _R. J. Mathar_, Jan 11 2020

%t r = x /. FindRoot[1/x + 1/(x+1)^2 == 1, {x, 2, 10}, WorkingPrecision -> 120]

%t RealDigits[r][[1]] (* A255249 *)

%t Table[Floor[n*r]], {n, 1, 250}] (* A330002 *)

%t Table[Floor[n*(1+r)^2], {n, 1, 250}] (* A330003 *)

%Y Cf. A329825, A255249, A330003 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 04 2020

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Last modified March 28 07:48 EDT 2024. Contains 371235 sequences. (Running on oeis4.)