A098501 -id:A098501 - cp's OEIS frontend

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

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

Ralf Stephan, Sep 15 2004

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

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

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

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

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

Ralf Stephan, Sep 15 2004

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

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Equals A098498(n) - A052938(n-4), n>3.

See A018836 (unbounded), A098498 (halfplane), A098500 (quadrant), A098501 (octant).

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

More terms from Colin Barker, Jul 14 2013

A098500 Number of squares on infinite quarter chessboard at <=n knight moves from the corner.

Original entry on oeis.org

1, 3, 12, 32, 59, 91, 130, 176, 229, 289, 356, 430, 511, 599, 694, 796, 905, 1021, 1144, 1274, 1411, 1555, 1706, 1864, 2029, 2201, 2380, 2566, 2759, 2959, 3166, 3380, 3601, 3829, 4064, 4306, 4555, 4811, 5074, 5344, 5621, 5905, 6196, 6494, 6799, 7111, 7430

Offset: 0

Ralf Stephan, Sep 15 2004

			3 squares are reachable after 1 move, from these you can reach 8 new squares more, so a(1)=3, a(2)=12.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences are in A047883.

See A018836 (unbounded), A098498 (halfplane), A098499 (diagonal halfplane), A098501 (octant).

a(n) = (1/2) * (7*n^2 + n + 2), for n>3.

G.f.: -(2*x^6-2*x^5-4*x^4+4*x^3+6*x^2+1) / (x-1)^3. - Colin Barker, Jul 15 2013

More terms from Colin Barker, Jul 15 2013

cp's OEIS Frontend

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

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

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A098500 Number of squares on infinite quarter chessboard at <=n knight moves from the corner.

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