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A186328 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 pentagonal numbers and the hexagonal numbers. Complement of A186329.

%I A186328 #6 Mar 30 2012 18:57:18
%S A186328 1,3,5,7,9,11,13,15,16,18,20,22,24,26,28,29,31,33,35,37,39,41,43,44,
%T A186328 46,48,50,52,54,56,57,59,61,63,65,67,69,71,72,74,76,78,80,82,84,85,87,
%U A186328 89,91,93,95,97,99,100,102,104,106,108,110,112,113,115,117,119,121,123,125,126,128,130,132,134,136,138,140,141,143,145,147,149,151,153,154,156,158,160,162,164,166,168,169,171,173,175,177,179,181,182,184,186
%N A186328 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 pentagonal numbers and the hexagonal numbers.  Complement of A186329.
%C A186328 See A186219 for a discussion of adjusted joint rank sequences.
%e A186328 First, write
%e A186328 1..5...12....22.....35......  (pentagonal)
%e A186328 1....6....15....28.......45.. (hexagonal)
%e A186328 Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the hexagonal:
%e A186328 a=(1,3,5,7,9,11,13,15,16,....)=A186328
%e A186328 b=(2,4,6,8,10,12,14,17,19,...)=A186329.
%t A186328 (* adjusted joint ranking; general formula *)
%t A186328 d=1/2; u=3/2; v=-1/2; w=0; x=2; y=-1; z=0;
%t A186328 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186328 a[n_]:=n+Floor[h[n]/(2x)];
%t A186328 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186328 b[n_]:=n+Floor[k[n]/(2u)];
%t A186328 Table[a[n], {n, 1, 100}]  (* A186328 *)
%t A186328 Table[b[n], {n, 1, 100}]  (* A186329 *)
%Y A186328 Cf. A186219, A186329, A186330, A186331,
%Y A186328 A000384 (pentagonal), A000384 (hexagonal).
%K A186328 nonn
%O A186328 1,2
%A A186328 _Clark Kimberling_, Feb 17 2011