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A352676 Intersection of Beatty sequences for sqrt(3) and 1+sqrt(3).

%I A352676 #16 Dec 26 2024 08:32:43
%S A352676 5,8,10,13,19,24,27,32,38,43,46,51,57,60,62,65,71,76,79,81,84,90,95,
%T A352676 98,103,109,112,114,117,122,128,131,133,136,142,147,150,152,155,161,
%U A352676 166,169,174,180,183,185,188,193,199,202,204,207,213,218,221,226,232
%N A352676 Intersection of Beatty sequences for sqrt(3) and 1+sqrt(3).
%C A352676 Conjectures:
%C A352676 (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 A352676 (2)  Limit_{n->oo} a(n)/n = (3/2)*(1+sqrt(3)).
%C A352676 (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 A352676 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 A352676 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%e A352676 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 A352676 z = 200; r = Sqrt[3]; s = 1 + Sqrt[3];
%t A352676 u = Table[Floor[n r], {n, 1, z}]    (* A022838 *)
%t A352676 v = Table[Floor[n s], {n, 1, z}]    (* A054088 *)
%t A352676 Intersection[u, v]  (* A352676 *)
%Y A352676 Cf. A022838, A054088.
%K A352676 nonn
%O A352676 1,1
%A A352676 _Clark Kimberling_, Mar 26 2022