cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A186513 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 A186514.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 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, 67, 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

Views

Author

Clark Kimberling, Feb 22 2011

Keywords

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).

Examples

			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 before 4+5j^2:
a=(1,2,3,5,7,8,9,11,12,14,15,17,..)=A186513
b=(4,6,10,13,16,19,22,26,29,32,...)=A186514.

Crossrefs

Cf. A186219, A186514, A186515, A186516.

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 + (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}]  (* A186513 *)
Table[b[n], {n, 1, 100}]  (* A186514 *)

Formula

a(n)=n+floor(sqrt((n^2)/5-9/10))=A186513(n).
b(n)=n+floor(sqrt(5n^2+9/2))=A186514(n).

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 22 2011

Keywords

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).

Examples

			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.

Crossrefs

Cf. A186219, A186513, A186514, A186516.

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 + (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 *)

Formula

a(n)=n+floor(sqrt((n^2)/5-7/10))=A186515(n).
b(n)=n+floor(sqrt(5n^2+7/2))=A186516(n).

A186516 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 A186515.

Original entry on oeis.org

3, 6, 9, 13, 16, 19, 22, 25, 29, 32, 35, 38, 42, 45, 48, 51, 55, 58, 61, 64, 67, 71, 74, 77, 80, 84, 87, 90, 93, 97, 100, 103, 106, 110, 113, 116, 119, 122, 126, 129, 132, 135, 139, 142, 145, 148, 152, 155, 158, 161, 165, 168, 171, 174, 177, 181, 184, 187, 190, 194, 197, 200, 203, 207, 210, 213, 216, 220, 223, 226, 229, 233, 236, 239, 242, 245, 249, 252, 255, 258, 262, 265, 268, 271
Offset: 1

Views

Author

Clark Kimberling, Feb 22 2011

Keywords

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).

Examples

			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.

Crossrefs

Cf. A186219, A186513, A186514, A186515.

Programs

Formula

a(n)=n+floor(sqrt((n^2)/5-7/10))=A186515(n).
b(n)=n+floor(sqrt(5n^2+7/2))=A186516(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.