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A186315 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 hexagonal numbers. Complement of A186316.

%I A186315 #5 Mar 30 2012 18:57:18
%S A186315 1,3,5,7,8,10,12,13,15,17,19,20,22,24,25,27,29,30,32,34,36,37,39,41,
%T A186315 42,44,46,48,49,51,53,54,56,58,59,61,63,65,66,68,70,71,73,75,77,78,80,
%U A186315 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 A186315 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 hexagonal numbers.  Complement of A186316.
%C A186315 See A186219 for a discussion of adjusted joint rank sequences.
%e A186315 First, write
%e A186315 1..4...9...16..25....36....49. (squares)
%e A186315 1....6...15.......28....45.... (hexagonal)
%e A186315 Replace each number by its rank, where ties are settled by ranking the square number before the hexagonal:
%e A186315 a=(1,3,5,7,8,10,12,13,...)=A186315.
%e A186315 b=(2,4,6,9,11,14,16,18,...)=A186316.
%t A186315 (* adjusted joint ranking; general formula *)
%t A186315 d=1/2; u=1; v=0; w=0; x=2; y=-1; z=0;
%t A186315 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186315 a[n_]:=n+Floor[h[n]/(2x)];
%t A186315 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186315 b[n_]:=n+Floor[k[n]/(2u)];
%t A186315 Table[a[n], {n, 1, 100}]  (* A186315 *)
%t A186315 Table[b[n], {n, 1, 100}]  (* A186316 *)
%Y A186315 Cf. A186219, A186316, A186317, A186318,
%Y A186315 A000290 (squares), A000384 (hexagonal numbers).
%K A186315 nonn
%O A186315 1,2
%A A186315 _Clark Kimberling_, Feb 17 2011