A186511 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-4+5j^2. Complement of A186512.

2, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 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, 110, 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

1,1

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-1),F(2h-1)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers).

Links

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

Formula

a(n)=n+floor(sqrt((n^2)/5+9/10))=A186511(n).

b(n)=n+floor(sqrt(5n^2-9/2))=A186512(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 after -4+5j^2:
a=(2,3,4,6,7,8,10,11,13,14,16,...)=A186511
b=(1,5,9,12,15,19,22,25,29,32,...)=A186512.

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 + (4 x (u*n^2 + v*n + w - z - d) + y^2)^(1/2);
a[n_] := n + Floor[h[n]/(2 x)];
k[n_] := -v + (4 u (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}]  (* A186511 *)
Table[b[n], {n, 1, 100}]  (* A186512 *)

Crossrefs

Cf. A186219, A186499, A186500, A186512.

Sequence in context: A354767 A351714 A248559 * A066049 A160697 A131629

Adjacent sequences: A186508 A186509 A186510 * A186512 A186513 A186514

Keyword

nonn

Author

Clark Kimberling, Feb 22 2011

Status

approved