A186389 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.
4, 6, 8, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92
Offset: 1
Keywords
Examples
First, write ......6.....12..18....24..30. (6i) 1..3..6..10...15....21..28... (triangular) Then replace each number by its rank, where ties are settled by ranking 6i after the triangular: a=(4,6,8,10,12,14,15,17,...)=A186389 b=(1,2,3,5,7,9,11,13,16,...)=A186390.
Programs
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Mathematica
(* 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 *) d=-1/2; u=6; v=0; x=1/2; y=1/2; (* 6i and triangular *) h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x); a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *) k[n_]:=(x*n^2+y*n-v+d)/u; b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *) Table[a[n], {n, 1, 120}] (* A186389 *) Table[b[n], {n, 1, 100}] (* A186390 *)
Comments