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A186541 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186542.

%I A186541 #4 Mar 30 2012 18:57:19
%S A186541 2,3,4,6,8,9,11,12,14,15,17,18,20,22,23,25,26,28,30,31,33,34,36,37,39,
%T A186541 41,42,44,45,47,48,50,52,53,55,56,58,59,61,63,64,66,67,69,70,72,74,75,
%U A186541 77,78,80,82,83,85,86,88,89,91,93,94,96,97,99,100,102,104,105,107,108,110,112,113,115,116,118,119,121,123,124,126,127,129,130,132,134,135,137,138,140,141,143,145,146,148,149,151,153,154,156,157
%N A186541 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2.  Complement of A186542.
%C A186541 See A186219 for a discussion of adjusted joint rank sequences.
%F A186541 a(n)=n+floor(sqrt((1/3)n^2+5/6))=A186541(n).
%F A186541 b(n)=n+floor(sqrt(3n^2-5/2))=A186542(n).
%e A186541 First, write
%e A186541 1..4..9..16..25..36..49..... (i^2)
%e A186541 .........10.....25.....46.. (-2+3j^2)
%e A186541 Then replace each number by its rank, where ties are settled by ranking i^2 after -2+3j^2:
%e A186541 a=(2,3,4,6,8,9,11,12,14,15,17,18,..)=A186541
%e A186541 b=(1,5,7,10,13,16,19,21,24,27,29...)=A186542.
%t A186541 (* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)
%t A186541 d = -1/2; u = 1; v = 0; w = 0; x = 3; y = 0; z = -2;
%t A186541 h[n_] := -y + (4 x (u*n^2 + v*n + w - z - d) + y^2)^(1/2);
%t A186541 a[n_] := n + Floor[h[n]/(2 x)];
%t A186541 k[n_] := -v + (4 u (x*n^2 + y*n + z - w + d) + v^2)^(1/2);
%t A186541 b[n_] := n + Floor[k[n]/(2 u)];
%t A186541 Table[a[n], {n, 1, 100}]  (* A186539 *)
%t A186541 Table[b[n], {n, 1, 100}]  (* A186540 *)
%Y A186541 Cf. A186219, A186539, A186540, A186542.
%K A186541 nonn
%O A186541 1,1
%A A186541 _Clark Kimberling_, Feb 23 2011