A186499 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-4+5j^2. Complement of A186500.

1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 109, 111, 112, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 133, 134, 136, 137, 138, 140, 141, 143, 144

Offset: 1

Author

Clark Kimberling, Feb 22 2011

Keywords

• nonn

Comments

See A186219 for a discussion of adjusted joint rank sequences.

The pairs (i,j) for which i^2=-4+5j^2 are (L(2h-2),F(2h-1)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers); compare this with the comment at A186511.

Examples

			First, write
1..4..9..16..25..36..49..... (i^2)
1........16........41........(-4+5j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -4+5j^2:
a=(1,3,4,5,7,8,10,11,13,14,15,17,18...)=A186499
b=(2,6,9,12,16,19,22,25,29,32,35,38,..)=A186500.
		

Crossrefs

Cf. A186219, A186500, A186511, A186512.

Programs

Mathematica

(* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)
d=1/2; u=1; v=0; w=0; x=5; y=0; z=4;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2 x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n + Floor[k[n]/(2 u)];
Table[a[n], {n, 1, 100}]  (* A186499 *)
Table[b[n], {n, 1, 100}]  (* A186500 *)

Formula

a(n)=n+floor((1/10)(sqrt(2n^2+7)))=A186499(n).

b(n)=n+floor(sqrt(5n^2-7/2))=A186500(n).