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

1,1

Links

Table of n, a(n) for n=1..120.

Example

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.

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 *)

Crossrefs

Cf. A186554, A186555, A186557.

Sequence in context: A354140 A183872 A288934 * A184582 A184734 A189662

Adjacent sequences: A186353 A186354 A186355 * A186357 A186358 A186359

Keyword

nonn

Author

Clark Kimberling, Feb 18 2011

Status

approved