A186383 - OEIS

%I #7 Mar 30 2012 18:57:18

%S 3,5,7,9,11,13,14,16,17,19,20,22,23,25,26,28,29,30,32,33,34,36,37,38,

%T 40,41,42,44,45,46,48,49,50,51,53,54,55,56,58,59,60,61,63,64,65,66,68,

%U 69,70,71,73,74,75,76,77,79,80,81,82,83,85,86,87,88,89,91

%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)=5i and g(j)=j(j+1)/2 (triangular number).  Complement of A186384.

%e First, write

%e .....5...10..15..20..25..30.. (5i)

%e 1..3..6..10..15....21..28.. (triangular)

%e Then replace each number by its rank, where ties are settled by ranking 5i before the triangular:

%e a=(3,5,7,9,11,13,14,16,17,..)=A186383

%e b=(1,2,4,6,8,10,12,15,18,...)=A186384.

%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=5; v=0; x=1/2; y=1/2; (* 5i 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}]  (* A186383 *)

%t Table[b[n], {n, 1, 100}]  (* A186384 *)

%Y Cf. A186350, A186384, A186385, A186386.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 19 2011