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%I #7 Mar 30 2012 18:57:18
%S 2,4,6,8,9,11,12,14,15,17,18,19,21,22,23,25,26,27,29,30,31,32,34,35,
%T 36,37,39,40,41,42,44,45,46,47,48,50,51,52,53,54,56,57,58,59,60,62,63,
%U 64,65,66,67,69,70,71,72,73,74,76,77,78,79,80,81,83,84,85,86,87,88,89,91,92,93,94,95,96,97,99,100,101,102,103,104,105,107,108,109,110,111,112,113,114,116,117,118,119,120,121,122,123,125,126,127,128,129,130,131,132,134,135
%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186355.
%C See A186350.
%e First, write
%e ...3..6..9....12..15..18..21..24.. (3*i)
%e 1..3..6....10.....15......21.... (triangular)
%e Then replace each number by its rank, where ties are settled by ranking 3i before the triangular:
%e a=(2,4,6,8,9,11,12,14,15,17,....)=A186354
%e b=(1,3,5,7,10,13,16,20,24,28,...)=A186355.
%t (* adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
%t d=1/2; u=3; v=0; x=1/2; y=1/2; (* odds and triangular *)
%t h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
%t a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *)
%t k[n_]:=(x*n^2+y*n-v+d)/u;
%t b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *)
%t Table[a[n],{n,1,120}] (* A186354 *)
%t Table[b[n],{n,1,100}] (* A186355 *)
%Y Cf. A186550, A186555, A186556, A186557.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 18 2011
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