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 1-4 of 4 results.

A098498 Number of squares on infinite half chessboard at <=n knight moves from a fixed point on the edge.

Original entry on oeis.org

1, 5, 23, 60, 110, 172, 248, 338, 442, 560, 692, 838, 998, 1172, 1360, 1562, 1778, 2008, 2252, 2510, 2782, 3068, 3368, 3682, 4010, 4352, 4708, 5078, 5462, 5860, 6272, 6698, 7138, 7592, 8060, 8542, 9038, 9548, 10072, 10610, 11162, 11728, 12308, 12902, 13510
Offset: 0

Views

Author

Ralf Stephan, Sep 15 2004

Keywords

Examples

			5 squares are reachable after 1 move, from these you can reach 18 new squares more, so a(1)=5 and a(2)=23.

Links

Crossrefs

See A018836 (unbounded), A098499 (diagonal halfplane), A098500 (quadrant), A098501 (octant).

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1}, {1, 5, 23, 60, 110, 172, 248}, 50] (* Paolo Xausa, Jul 17 2024 *)

Formula

a(n) = 7*n^2 - n + 2, for n>3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>6. G.f.: -(2*x^6 -x^5 -6*x^4 +5*x^3 +11*x^2 +2*x +1) / (x -1)^3. - Colin Barker, Jul 14 2013

Extensions

More terms from Colin Barker, Jul 14 2013

A098499 Number of squares on infinite half chessboard at <=n knight moves from a fixed point on the diagonal.

Original entry on oeis.org

1, 5, 23, 57, 109, 169, 246, 334, 439, 555, 688, 832, 993, 1165, 1354, 1554, 1771, 1999, 2244, 2500, 2773, 3057, 3358, 3670, 3999, 4339, 4696, 5064, 5449, 5845, 6258, 6682, 7123, 7575, 8044, 8524, 9021, 9529, 10054, 10590, 11143, 11707, 12288, 12880, 13489
Offset: 0

Views

Author

Ralf Stephan, Sep 15 2004

Keywords

Examples

			5 squares are reachable after 1 move, from these you can reach 18 new squares more, so a(1)=5, a(2)=23.

Links

Crossrefs

Equals A098498(n) - A052938(n-4), n>3.
See A018836 (unbounded), A098498 (halfplane), A098500 (quadrant), A098501 (octant).

Formula

a(n) = (1/4) [28n^2 - 6n + 9 + 3(-1)^n], for n>3.
G.f.: -(3*x^7-x^6-8*x^5+4*x^4+13*x^3+13*x^2+3*x+1) / ((x-1)^3*(x+1)). - Colin Barker, Jul 14 2013

Extensions

More terms from Colin Barker, Jul 14 2013

A098501 Number of squares on infinite octant of chessboard at <=n knight moves from the corner. The octant includes the diagonal.

Original entry on oeis.org

1, 2, 5, 13, 31, 49, 70, 93, 121, 151, 186, 223, 265, 309, 358, 409, 465, 523, 586, 651, 721, 793, 870, 949, 1033, 1119, 1210, 1303, 1401, 1501, 1606, 1713, 1825, 1939, 2058, 2179, 2305, 2433, 2566, 2701, 2841, 2983, 3130, 3279, 3433, 3589, 3750, 3913, 4081
Offset: 0

Views

Author

Ralf Stephan, Sep 15 2004

Keywords

Examples

			2 squares are reachable after 1 move, from these you can reach 3 new squares more, so a(1)=2, a(2)=5.

Links

Crossrefs

See A018836 (unbounded), A098498 (halfplane), A098499 (diagonal halfplane), A098500 (quadrant).

Formula

a(n) = (1/8) * [14n^2 + 8n + 5 + 3(-1)^n], for n>4.
G.f.: -(2*x^8+2*x^7-7*x^6-5*x^5+8*x^4+5*x^3+x^2+1) / ((x-1)^3*(x+1)). - Colin Barker, Jul 14 2013

Extensions

More terms from Colin Barker, Jul 14 2013

A047883 Squares on unbounded chessboard for which the least number of knight's moves from corner (0,0) is n.

Original entry on oeis.org

0, 2, 9, 20, 27, 32, 39, 46, 53, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 130, 137, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 214, 221, 228, 235
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A098500 (partial sums).

Programs

  • Mathematica
    Join[{0,2,9,20,27},NestList[#+7&,32,30]] (* Harvey P. Dale, Sep 07 2013 *)

Formula

a(n)=7+a(n-1) for n >= 6.
Showing 1-4 of 4 results.

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