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A186320 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the squares and heptagonal numbers. Complement of A186321.

%I A186320 #5 Mar 30 2012 18:57:18
%S A186320 1,3,5,6,8,10,11,13,14,16,18,19,21,23,24,26,28,29,31,32,34,36,37,39,
%T A186320 41,42,44,46,47,49,50,52,54,55,57,59,60,62,63,65,67,68,70,72,73,75,77,
%U A186320 78,80,81,83,85,86,88,90,91,93,94,96,98,99,101,103,104,106,108,109,111,112,114,116,117,119,121,122,124,125,127,129,130,132,134,135,137,139,140,142,143,145,147,148,150,152,153,155,157,158,160,161,163
%N A186320 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the squares and heptagonal numbers.  Complement of A186321.
%e A186320 First, write
%e A186320 1..4...9..16....25...36...49...64.. (squares)
%e A186320 1....7.......18....34........55.... (heptagonal)
%e A186320 Replace each number by its rank, where ties are settled by ranking the square number before the heptagonal:
%e A186320 a=(1,3,5,6,8,10,11,13,...)=A186320
%e A186320 b=(2,4,7,9,12,15,17,20,...)=A186321.
%t A186320 (* adjusted joint ranking; general formula *)
%t A186320 d=1/2; u=1; v=0; w=0; x=5/2; y=-3/2; z=0;
%t A186320 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186320 a[n_]:=n+Floor[h[n]/(2x)];
%t A186320 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186320 b[n_]:=n+Floor[k[n]/(2u)];
%t A186320 Table[a[n], {n, 1, 100}]  (* A186320 *)
%t A186320 Table[b[n], {n, 1, 100}]  (* A186321 *)
%Y A186320 Cf. A186219, A186321, A186322, A186323,
%Y A186320 A000290 (squares), A000566 (heptagonal numbers).
%K A186320 nonn
%O A186320 1,2
%A A186320 _Clark Kimberling_, Feb 17 2011