A186356 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.
3, 5, 6, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 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
Offset: 1
Keywords
Examples
First, write ...3..6..9....12..15..18..21..24.. (3*i) 1..3..6....10.....15......21.... (triangular) Then replace each number by its rank, where ties are settled by ranking 3i after the triangular: a=(3,5,6,8,10,11,13,14,15,..)=A186356 b=(1,2,4,7,9,12,16,19,23,...)=A186357.
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=3; v=0; x=1/2; y=1/2; 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}] (* A186356 *) Table[b[n],{n,1,100}] (* A186357 *)