A186379 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186380.

3, 5, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88

Offset: 1

Author

Clark Kimberling, Feb 19 2011

Keywords

• nonn

Comments

See A186350.

Examples

			First, write
.....4..8..12..16..20..24..28.. (4*i)
1..3..6..10..15.....21.....28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 4i before the triangular:
a=(3,5,7,9,10,12,13,15,16,..)=A186379
b=(1,2,4,6,8,11,14,17,20,...)=A186380.
		

Crossrefs

Cf. A186350, A186380, A186381, A186382.

Programs

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=4; v=0; x=1/2; y=1/2; (* 4i 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}]  (* A186379 *)
Table[b[n], {n, 1, 100}]  (* A186380 *)

Formula

a(n)=n+floor(-1/2+sqrt(8n-3/4))=A186379(n).

b(n)=n+floor((n^2+n+1)/8)=A186380(n).