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A186159 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 triangular numbers and octagonal numbers. Complement of A186274.

%I A186159 #7 Mar 30 2012 18:57:18
%S A186159 1,3,4,6,7,8,10,11,13,14,16,17,18,20,21,23,24,25,27,28,30,31,32,34,35,
%T A186159 37,38,39,41,42,44,45,47,48,49,51,52,54,55,56,58,59,61,62,63,65,66,68,
%U A186159 69,70,72,73,75,76,77,79,80,82,83,85,86,87,89,90,92,93,94,96,97,99,100,101,103,104,106,107,108,110,111,113,114,116,117,118,120,121,123,124,125,127,128,130,131,132,134,135,137,138,139,141
%N A186159 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 triangular numbers and octagonal numbers.  Complement of A186274.
%C A186159 See A186219 for a discussion of adjusted joint rank sequences.
%e A186159 First, write the triangular and octagonal numbers:
%e A186159 1..3..6.....10..15..21..28
%e A186159 1........8..........21......
%e A186159 Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal:
%e A186159 a=(1,3,4,6,7,8,10,11,13,...)=A186159.
%e A186159 b=(2,5,9,12,15,19,22,26,...)=A186274.
%t A186159 (* adjusted joint ranking; general formula *)
%t A186159 d=1/2; u=1/2; v=1/2; w=0; x=3; y=-2; z=0;
%t A186159 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186159 a[n_]:=n+Floor[h[n]/(2x)];
%t A186159 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186159 b[n_]:=n+Floor[k[n]/(2u)];
%t A186159 Table[a[n],{n,1,100}] (* A186159 *)
%t A186159 Table[b[n],{n,1,100}] (* A186274 *)
%Y A186159 Cf. A186219, A186274, A186275, A186276,
%Y A186159 Cf. A000217 (triangular numbers).
%Y A186159 Cf. A000567 (octagonal numbers).
%K A186159 nonn
%O A186159 1,2
%A A186159 _Clark Kimberling_, Feb 13 2011