A186384 - OEIS

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

%S 1,2,4,6,8,10,12,15,18,21,24,27,31,35,39,43,47,52,57,62,67,72,78,84,

%T 90,96,102,109,116,123,130,137,145,153,161,169,177,186,195,204,213,

%U 222,232,242,252,262,272,283,294,305,316,327,339,351,363,375,387,400,413,426,439,452,466,480,494,508,522,537,552,567,582,597,613,629,645,661,677,694,711,728,745,762,780,798,816,834,852,871,890

%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 A186383.

%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, A186383, A186385, A186386.

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 19 2011