|
%I #6 Mar 30 2012 18:57:18
%S 3,5,7,9,10,12,14,15,17,18,19,21,22,24,25,26,28,29,30,32,33,34,36,37,
%T 38,39,41,42,43,45,46,47,48,50,51,52,53,54,56,57,58,59,61,62,63,64,65,
%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) after g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186382.
%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 after the triangular:
%e a=(3,5,7,9,10,12,14,15,17,..)=A186381
%e b=(1,2,4,6,8,11,13,16,20,...)=A186382.
%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}] (* A186381 *)
%t Table[b[n], {n, 1, 100}] (* A186382 *)
%Y Cf. A186379, A186380, A186382.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 19 2011
|
