A186356 - OEIS

%I #4 Mar 30 2012 18:57:18

%S 3,5,6,8,10,11,13,14,15,17,18,20,21,22,24,25,26,27,29,30,31,33,34,35,

%T 36,38,39,40,41,42,44,45,46,47,49,50,51,52,53,55,56,57,58,59,60,62,63,

%U 64,65,66,68,69,70,71,72,73,75,76,77,78,79,80,81,83,84,85,86,87,88,90,91,92,93,94,95,96,98,99,100,101,102,103,104,105,107,108,109,110,111,112,113,115,116,117,118,119,120,121,122,124,125,126,127,128,129,130,131,132,134,135,136,137,138,139,140,141,143,144,145,146

%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)=3i and g(j)=j(j+1)/2 (triangular number).  Complement of A186357.

%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 after the triangular:

%e a=(3,5,6,8,10,11,13,14,15,..)=A186356

%e b=(1,2,4,7,9,12,16,19,23,...)=A186357.

%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;

%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}]  (* A186356 *)

%t Table[b[n],{n,1,100}]  (* A186357 *)

%Y Cf. A186554, A186555, A186557.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 18 2011