A186354 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186355.

2, 4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135

Offset

1,1

Comments

See A186350.

Links

Table of n, a(n) for n=1..110.

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 before the triangular:
a=(2,4,6,8,9,11,12,14,15,17,....)=A186354
b=(1,3,5,7,10,13,16,20,24,28,...)=A186355.

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; (* odds 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}]  (* A186354 *)
Table[b[n], {n, 1, 100}]  (* A186355 *)

Crossrefs

Cf. A186550, A186555, A186556, A186557.

Sequence in context: A191982 A080037 A184012 * A186149 A298861 A114571

Adjacent sequences: A186351 A186352 A186353 * A186355 A186356 A186357

Keyword

nonn

Author

Clark Kimberling, Feb 18 2011

Status

approved