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%I #4 Mar 30 2012 18:57:18
%S 2,3,5,6,8,10,11,13,15,16,18,19,21,23,24,26,28,29,31,32,34,36,37,39,
%T 41,42,44,46,47,49,50,52,54,55,57,59,60,62,63,65,67,68,70,72,73,75,77,
%U 78,80,81,83,85,86,88,90,91,93,94,96,98,99,101,103,104,106,108,109,111,112,114,116,117,119,121,122,124,126,127,129,130,132,134,135,137,139,140,142,143,145,147,148,150,152,153,155,157,158,160,161,163
%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 heptagonal numbers. Complement of A186323.
%e First, write
%e 1..4...9..16....25...36...49...64.. (squares)
%e 1....7.......18....34........55.... (heptagonal)
%e Replace each number by its rank, where ties are settled by ranking the square number after the heptagonal:
%e a=(2,3,5,6,8,10,11,13,...)=A186322
%e b=(1,4,7,9,12,15,17,20,...)=A186323.
%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}] (* A186322 *)
%t Table[b[n], {n, 1, 100}] (* A186323 *)
%Y Cf. A186219, A186320, A186321, A186323.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 17 2011
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