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A186539 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186540. 4

%I #10 May 18 2020 12:05:56

%S 1,3,4,6,7,9,11,12,14,15,17,18,20,22,23,25,26,28,29,31,33,34,36,37,39,

%T 41,42,44,45,47,48,50,52,53,55,56,58,59,61,63,64,66,67,69,70,72,74,75,

%U 77,78,80,82,83,85,86,88,89,91,93,94,96,97,99,100,102,104,105,107,108,110,111,113,115,116,118,119,121,123,124,126,127,129,130,132,134,135,137,138,140,141,143,145,146,148,149,151,153,154,156,157

%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186540.

%C See A186219 for a discussion of adjusted joint rank sequences.

%C Differs from A059555 at n=97, 123, 194, 220, 246, ... - _R. J. Mathar_, May 18 2020

%F a(n)=n+floor(sqrt((1/3)n^2+1/24))=A186539(n).

%F b(n)=n+floor(sqrt(3n^2-3/2))=A186540(n).

%e First, write

%e 1..4..9..16..25..36..49.... (i^2)

%e .......10....25.....46.. (-2+3j^2)

%e Then replace each number by its rank, where ties are settled by ranking i^2 before -2+3j^2:

%e a=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539

%e b=(2,5,8,10,13,16,19,21,24,27,30...)=A186540.

%t (* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)

%t d = 1/2; u = 1; v = 0; w = 0; x = 3; y = 0; z = -2;

%t h[n_] := -y + (4 x (u*n^2 + v*n + w - z - d) + y^2)^(1/2);

%t a[n_] := n + Floor[h[n]/(2 x)];

%t k[n_] := -v + (4 u (x*n^2 + y*n + z - w + d) + v^2)^(1/2);

%t b[n_] := n + Floor[k[n]/(2 u)];

%t Table[a[n], {n, 1, 100}] (* A186539 *)

%t Table[b[n], {n, 1, 100}] (* A186540 *)

%Y Cf. A186219, A186540, A186541, A186542.

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 23 2011

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