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%I #7 Mar 30 2012 18:57:18
%S 3,5,7,9,10,12,13,15,16,18,19,21,22,24,25,26,28,29,30,32,33,34,36,37,
%T 38,39,41,42,43,44,46,47,48,49,51,52,53,54,56,57,58,59,61,62,63,64,65,
%U 67,68,69,70,71,73,74,75,76,77,79,80,81,82,83,84,86,87,88
%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)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186380.
%C See A186350.
%F a(n)=n+floor(-1/2+sqrt(8n-3/4))=A186379(n).
%F b(n)=n+floor((n^2+n+1)/8)=A186380(n).
%e First, write
%e .....4..8..12..16..20..24..28.. (4*i)
%e 1..3..6..10..15.....21.....28.. (triangular)
%e Then replace each number by its rank, where ties are settled by ranking 4i before the triangular:
%e a=(3,5,7,9,10,12,13,15,16,..)=A186379
%e b=(1,2,4,6,8,11,14,17,20,...)=A186380.
%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=4; v=0; x=1/2; y=1/2; (* 4i 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}] (* A186379 *)
%t Table[b[n], {n, 1, 100}] (* A186380 *)
%Y Cf. A186350, A186380, A186381, A186382.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 19 2011
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