|
%I #11 May 03 2022 23:52:32
%S 5,8,10,13,19,24,27,32,38,43,46,51,57,60,62,65,71,76,79,81,84,90,95,
%T 98,103,109,112,114,117,122,128,131,133,136,142,147,150,152,155,161,
%U 166,169,174,180,183,185,188,193,199,202,204,207,213,218,221,226,232
%N Intersection of Beatty sequences for sqrt(3) and 1+sqrt(3).
%C Conjectures:
%C (1) a(n+1)-a(n) is in (2,3,4,5,6} for every n, and each of these differences occurs infinitely many times.
%C (2) Limit_{n->oo} a(n)/n = (3/2)*(1+sqrt(3)).
%C (3) Let d(n) = a(n) - A352673(n); then d(n) = 0 for infinitely many n, but {d(n)} is unbounded below and above.
%H Clark Kimberling, R. Stanley, A. Kalmynin, <a href="https://mathoverflow.net/questions/418749/limit-associated-with-two-beatty-sequences-that-are-not-a-beatty-pair"> Limit associated with two Beatty sequences that are not a Beatty pair</a>, Math Overflow, Mar 2022.
%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%e The two Beatty sequences, (1,3,5,6,8,10,12,13,15,17,19,20,...) and (2,5,8,10,13,16,19,21,24,...), share the numbers (5,8,10,13,19,24,...).
%t z = 200; r = Sqrt[3]; s = 1 + Sqrt[3];
%t u = Table[Floor[n r], {n, 1, z}] (* A022838 *)
%t v = Table[Floor[n s], {n, 1, z}] (* A054088 *)
%t Intersection[u, v] (* A352676 *)
%Y Cf. A022838, A054088.
%K nonn
%O 1,1
%A _Clark Kimberling_, Mar 26 2022
|
