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.

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.

Original entry on oeis.org

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

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

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 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.

Crossrefs

Cf. A186219, A186499, A186500, 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 + (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 *)

Formula

a(n)=n+floor(sqrt((n^2)/5+9/10))=A186511(n).
b(n)=n+floor(sqrt(5n^2-9/2))=A186512(n).

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.

Original entry on oeis.org

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

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-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).

A186500 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 A186499.

Original entry on oeis.org

2, 6, 9, 12, 16, 19, 22, 25, 29, 32, 35, 38, 42, 45, 48, 51, 54, 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, 232, 236, 239, 242, 245, 249, 252, 255, 258, 262, 265
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-2),F(2h-1)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers).

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, A186499, A186511, A186512.

Programs

Formula

a(n)=n+floor((1/10)(sqrt(2n^2+7)))=A186499(n).
b(n)=n+floor(sqrt(5n^2-7/2))=A186500(n).
Showing 1-3 of 3 results.

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