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

1,2

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.

Links

Table of n, a(n) for n=1..100.

Formula

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

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

Example

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.

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 *)

Crossrefs

Cf. A186219, A186500, A186511, A186512.

Sequence in context: A136119 A184618 A110882 * A187911 A089230 A171512

Adjacent sequences: A186496 A186497 A186498 * A186500 A186501 A186502

Keyword

nonn

Author

Clark Kimberling, Feb 22 2011

Status

approved