A186315 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the squares and hexagonal numbers. Complement of A186316.

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111, 112, 114, 116, 118, 119, 121, 123, 124, 126, 128, 129, 131, 133, 135, 136, 138, 140, 141, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 170

Offset: 1

Author

Clark Kimberling, Feb 17 2011

Keywords

• nonn

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..4...9...16..25....36....49. (squares)
1....6...15.......28....45.... (hexagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the hexagonal:
a=(1,3,5,7,8,10,12,13,...)=A186315.
b=(2,4,6,9,11,14,16,18,...)=A186316.
		

Crossrefs

Cf. A186219, A186316, A186317, A186318,

A000290 (squares), A000384 (hexagonal numbers).

Programs

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=2; y=-1; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186315 *)
Table[b[n], {n, 1, 100}]  (* A186316 *)