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Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and hexagonal numbers. Complement of A186318.
4

%I #4 Mar 30 2012 18:57:18

%S 2,3,5,7,8,10,12,13,15,17,19,20,22,24,25,27,29,30,32,34,36,37,39,41,

%T 42,44,46,48,49,51,53,54,56,58,60,61,63,65,66,68,70,71,73,75,77,78,80,

%U 82,83,85,87,89,90,92,94,95,97,99,100,102,104,106,107,109,111,112,114,116,118,119,121,123,124,126,128,129,131,133,135,136,138,140,141,143,145,147,148,150,152,153,155,157,159,160,162,164,165,167,169,170

%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and hexagonal numbers. Complement of A186318.

%e First, write

%e 1..4...9...16..25....36....49. (squares)

%e 1....6...15.......28....45.... (hexagonals)

%e Replace each number by its rank, where ties are settled by ranking the square number after the hexagonal:

%e a=(2,3,5,7,8,10,12,13,...)=A186317.

%e b=(1,4,6,9,11,14,16,18,...)=A186318.

%t (* adjusted joint ranking; general formula *)

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

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

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

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

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

%t Table[a[n], {n, 1, 100}] (* A186317 *)

%t Table[b[n], {n, 1, 100}] (* A186318 *)

%Y Cf. A186315, A186316, A186318.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 17 2011