A186227 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 triangular numbers and heptagonal numbers. Complement of A186228.

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 109, 110, 111, 113, 114, 116, 117, 119, 120, 122, 123, 124, 126, 127, 129, 130, 132, 133, 135, 136, 138, 139, 140, 142, 143, 145

Offset: 1

Author

Clark Kimberling, Feb 16 2011

Keywords

• nonn

Comments

See A186219 for a general discussion of adjusted joint rank sequences.

Examples

			First, write
1..3..6..10..15..21..28..36..45... (triangular)
1.......7......18......34.......55... (heptagonal)
Then replace each number by its rank, where ties are settled by ranking the triangular number before the heptagonal:
a=(1,3,4,6,7,9,10,12,...), A186227.
b=(2,5,8,11,15,18,21,...), A186228.
		

Crossrefs

Cf. A186219, A186228, A186237, A186238,

Cf. A000217 (triangular numbers)

Cf. A000566 (heptagonal numbers)

Programs

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1/2; v=1/2; w=0; x=5/2; y=-3/2; 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}]  (* A186227 *)
Table[b[n],{n,1,100}]  (* A186228 *)