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.

A186539 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)=-2+3j^2. Complement of A186540.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108, 110, 111, 113, 115, 116, 118, 119, 121, 123, 124, 126, 127, 129, 130, 132, 134, 135, 137, 138, 140, 141, 143, 145, 146, 148, 149, 151, 153, 154, 156, 157
Offset: 1

Views

Author

Clark Kimberling, Feb 23 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.
Differs from A059555 at n=97, 123, 194, 220, 246, ... - R. J. Mathar, May 18 2020

Examples

			First, write
1..4..9..16..25..36..49.... (i^2)
.......10....25.....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -2+3j^2:
a=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539
b=(2,5,8,10,13,16,19,21,24,27,30...)=A186540.

Crossrefs

Cf. A186219, A186540, A186541, A186542.

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 = 3; y = 0; z = -2;
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}]  (* A186539 *)
Table[b[n], {n, 1, 100}]  (* A186540 *)

Formula

a(n)=n+floor(sqrt((1/3)n^2+1/24))=A186539(n).
b(n)=n+floor(sqrt(3n^2-3/2))=A186540(n).

A186541 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)=-2+3j^2. Complement of A186542.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108, 110, 112, 113, 115, 116, 118, 119, 121, 123, 124, 126, 127, 129, 130, 132, 134, 135, 137, 138, 140, 141, 143, 145, 146, 148, 149, 151, 153, 154, 156, 157
Offset: 1

Views

Author

Clark Kimberling, Feb 23 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..4..9..16..25..36..49..... (i^2)
.........10.....25.....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 after -2+3j^2:
a=(2,3,4,6,8,9,11,12,14,15,17,18,..)=A186541
b=(1,5,7,10,13,16,19,21,24,27,29...)=A186542.

Crossrefs

Cf. A186219, A186539, A186540, A186542.

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 = 3; y = 0; z = -2;
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}]  (* A186539 *)
Table[b[n], {n, 1, 100}]  (* A186540 *)

Formula

a(n)=n+floor(sqrt((1/3)n^2+5/6))=A186541(n).
b(n)=n+floor(sqrt(3n^2-5/2))=A186542(n).

A186542 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)=-2+3j^2. Complement of A186541.

Original entry on oeis.org

1, 5, 7, 10, 13, 16, 19, 21, 24, 27, 29, 32, 35, 38, 40, 43, 46, 49, 51, 54, 57, 60, 62, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 111, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 152, 155, 158, 161, 163, 166, 169, 172, 174, 177, 180, 183, 185, 188, 191, 193, 196, 199, 202, 204, 207, 210, 213, 215, 218, 221, 224, 226, 229, 232, 234, 237, 240
Offset: 1

Views

Author

Clark Kimberling, Feb 23 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..4..9..16..25..36..49..... (i^2)
.........10.....25.....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 after -2+3j^2:
a=(2,3,4,6,8,9,11,12,14,15,17,18,..)=A186541
b=(1,5,7,10,13,16,19,21,24,27,29...)=A186542.

Links

Crossrefs

Cf. A186219, A186539, A186540, A186541.

Programs

Formula

a(n)=n+floor(sqrt((1/3)n^2+5/6))=A186541(n).
b(n)=n+floor(sqrt(3n^2-5/2))=A186542(n).
Showing 1-3 of 3 results.

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