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A186497 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-2 and g(j)=j-th triangular number. Complement of A186498.

%I A186497 #10 Apr 09 2015 15:02:38
%S A186497 1,4,6,7,9,11,12,14,15,16,18,19,21,22,23,25,26,27,28,30,31,32,34,35,
%T A186497 36,37,39,40,41,42,43,45,46,47,48,50,51,52,53,54,56,57,58,59,60,61,63,
%U A186497 64,65,66,67,69,70,71,72,73,74,76,77,78,79,80,81,82,84,85,86,87,88,89,91,92,93,94,95,96,97,99,100,101,102,103,104,105,106,108,109,110,111,112,113,114,116,117,118,119,120,121,122,123,125,126,127,128,129,130,131,132,133,135,136,137,138,139,140,141,142,144,145,146
%N A186497 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-2 and g(j)=j-th triangular number.  Complement of A186498.
%C A186497 See A186350 for a discussion of adjusted joint rank sequences.
%e A186497 First, write
%e A186497 1..4..7.10..13..16..19..22..25..28..31. (3i-2),
%e A186497 1.3..6..10....15.......21.......28.....(j(j+1)/2).
%e A186497 Then replace each number by its rank, where ties are settled by ranking 3i-2 before j(j+1)/2:
%e A186497 a=(1,4,6,7,9,11,12,14,15,16,18,...)=A186497,
%e A186497 b=(2,3,5,8,10,13,17,20,24,29,33,..)=A186498.
%t A186497 (* 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 A186497 d=1/2; u=3; v=-2; x=1/2; y=1/2;
%t A186497 h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
%t A186497 a[n_]:=n+Floor[h[n]];
%t A186497 k[n_]:=(x*n^2+y*n-v+d)/u;
%t A186497 b[n_]:=n+Floor[k[n]];
%t A186497 Table[a[n],{n,1,120}]  (* A186497 *)
%t A186497 Table[b[n],{n,1,100}]  (* A186498 *)
%Y A186497 Cf. A186350, A186498, A186499, A186500.
%K A186497 nonn
%O A186497 1,2
%A A186497 _Clark Kimberling_, Feb 22 2011