A186515 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 A186516.

1, 2, 4, 5, 7, 8, 10, 11, 12, 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, 54, 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),F(2h)), 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)/5-7/10))=A186515(n).

b(n)=n+floor(sqrt(5n^2+7/2))=A186516(n).

Example

First, write
1..4..9..16..25..36..49..... (i^2)
......9.....24.......49.. (4+5j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 after 4+5j^2:
a=(1,2,4,5,7,8,10,11,12,14,15,17,..)=A186515
b=(3,6,9,13,16,19,22,25,29,32,35,..)=A186516.

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}]  (* A186515 *)
Table[b[n], {n, 1, 100}]  (* A186516 *)

Crossrefs

Cf. A186219, A186513, A186514, A186516.

Sequence in context: A307913 A059539 A307513 * A078934 A026496 A056127

Adjacent sequences: A186512 A186513 A186514 * A186516 A186517 A186518

Keyword

nonn

Author

Clark Kimberling, Feb 22 2011

Status

approved