A186352 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the odd numbers and the triangular numbers. Complement of A186353.

2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141

Offset: 1

Author

Clark Kimberling, Feb 18 2011

Keywords

• nonn

Examples

			First, write
1..3..5..7..9..11..13..15..17..21..23.. (odds)
1..3....6.....10.......15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking the odd number after the triangular:
a=(2,4,5,7,8,10,11,13,14,15,....)=A186352
b=(1,3,6,9,12,16,21,26,31,37,...)=A186353.
		

Crossrefs

Cf. A186350, A186351, A186353.

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=2; v=-1; 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}]  (* A186352 *)
Table[b[n], {n, 1, 100}]  (* A186353 *)

Formula

a(n)=n+floor(-1/2+sqrt(4n-3/4))=A186352(n).

b(n)=n+floor((n^2+n+1)/4)=A186353(n).