A186223 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 pentagonal numbers. Complement of A186224.

1, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158

Offset: 1

Author

Clark Kimberling, Feb 15 2011

Keywords

• nonn

Comments

See A186219 for a general description.

Examples

			First, write
1..3...6..10....15...21.....28......36...45...  (triangular)
1....5.........12...........22......35........... (pentagonal)
Replace each number by its rank, where ties are settled by ranking the triangular number before the pentagonal:
a=(1,3,5,6,8,9,11,13,...)
b=(2,4,7,10,12,15,18,...).
		

Crossrefs

Cf. A186219, A186224, A186225, A186226,

A000217 (triangular), A000326 (pentagonal).

Programs

Mathematica

d=1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0;
(* triangular & pentagonal *)
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}]  (* A186223 *)
Table[b[n],{n,1,100}]  (* A186224 *)