A186288 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 pentagonal numbers. Complement of A186289.

1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 120, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 150, 152, 154, 156, 158, 160, 161, 163, 165, 167, 169, 170, 172, 174, 176, 178, 179, 181

Offset: 1

Author

Clark Kimberling, Feb 17 2011

Keywords

• nonn

Comments

See A186219 for a discussion of adjusted rank sequences.

Examples

			First, write
1..4...9....16....25..36..49..... (squares)
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the pentagonal:
a=(1,3,5,7,9,11,12,14,....)=A186288.
b=(2,4,6,8,10,13,15,17,...)=A186289.
		

Crossrefs

Cf. A186219, A186289, A186290, A186291, A000290 (squares), A000326 (pentagonal).

Programs

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=3/2; y=-1/2; 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}]  (* A186288 *)
Table[b[n],{n,1,100}]  (* A186289 *)