A186389 - OEIS

%I #6 Mar 30 2012 18:57:18

%S 4,6,8,10,12,14,15,17,18,20,22,23,25,26,27,29,30,32,33,35,36,37,39,40,

%T 41,43,44,45,47,48,49,51,52,53,55,56,57,58,60,61,62,63,65,66,67,69,70,

%U 71,72,74,75,76,77,78,80,81,82,83,85,86,87,88,90,91,92

%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number).  Complement of A186390.

%C See A186350 for a discussion of adjusted joint rank sequences.

%e First, write

%e ......6.....12..18....24..30. (6i)

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

%e Then replace each number by its rank, where ties are settled by ranking 6i after the triangular:

%e a=(4,6,8,10,12,14,15,17,...)=A186389

%e b=(1,2,3,5,7,9,11,13,16,...)=A186390.

%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=6; v=0; x=1/2; y=1/2; (* 6i 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}]  (* A186389 *)

%t Table[b[n], {n, 1, 100}]  (* A186390 *)

%Y Cf. A186350, A186387, A186388, A186390.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 19 2011