A018836 -id:A018836 - 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

A297740 The number of distinct positions on an infinite chessboard reachable by the (2,3)-leaper in <= n moves.

Original entry on oeis.org

1, 9, 41, 129, 321, 625, 997, 1413, 1885, 2425, 3033, 3709, 4453, 5265, 6145, 7093, 8109, 9193, 10345, 11565, 12853, 14209, 15633, 17125, 18685, 20313, 22009, 23773, 25605, 27505, 29473, 31509, 33613, 35785, 38025, 40333, 42709, 45153, 47665, 50245, 52893

Offset: 0

R. J. Mathar, Jan 05 2018

Colin Barker, Table of n, a(n) for n = 0..1000
R. J. Mathar, Fairy chess leaper minimum moves on the infinite chessboard, (2018).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A018836 (1,2)-leaper or (1,3)-leaper, A297741 (3,4)-leaper.
Partial sums of A018839.
Cf. A253974, A254129, A254459.

LinearRecurrence[{3, -3, 1}, {1, 9, 41, 129, 321, 625, 997, 1413, 1885, 2425}, 50] (* Paolo Xausa, Mar 17 2024 *)

Vec((1 + x)*(1 + 5*x + 12*x^2 + 20*x^3 + 28*x^4 - 20*x^5 - 24*x^6 + 12*x^8) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Jan 07 2018

a(n) = 34*n^2 + 30*n + 9 for n >= 6.
From Colin Barker, Jan 05 2018: (Start)
G.f.: (1 + x)*(1 + 5*x + 12*x^2 + 20*x^3 + 28*x^4 - 20*x^5 - 24*x^6 + 12*x^8) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>9. (End)

A018842 Number of squares on infinite chessboard at n knight's moves from center.

Original entry on oeis.org

1, 8, 32, 68, 96, 120, 148, 176, 204, 232, 260, 288, 316, 344, 372, 400, 428, 456, 484, 512, 540, 568, 596, 624, 652, 680, 708, 736, 764, 792, 820, 848, 876, 904, 932, 960, 988, 1016, 1044, 1072, 1100, 1128, 1156

Offset: 0

N. J. A. Sloane, Marc LeBrun

Moon Duchin, Counting in Groups: Fine Asymptotic Geometry, Notices of the AMS 63.8 (2016), pp. 871-974. See p. 873.
Mordechai Katzman, Knight's moves on an infinite board
M. Katzman, Counting Monomials, J. Alg. Comb. 22 (2005) 331-341.
A. M. Miller and D. L. Farnsworth, Counting the Number of Squares Reachable in k Knight's Moves, Open Journal of Discrete Mathematics, 2013, 3, 151-154.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A018836 (partial sums), A038522.

(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^2; seq(coeff(series(%, x, n+1), x, n), n=0..50);

CoefficientList[Series[(1+5x+12x^2-8x^4+4x^5)(1+x)/(1-x)^2, {x,0,50}], x] (* or *) Join[{1,8,32,68,96},LinearRecurrence[{2,-1},{120,148},46]] (* Harvey P. Dale, Jul 05 2011 *)

a(n) = 28*n-20, n >= 5.

G.f.: (1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^2.

Formula corrected by Jean Drabbe, Mar 11 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

Ralf Stephan, Sep 15 2004

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

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

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

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

More terms from Colin Barker, Jul 14 2013

A297741 The number of distinct positions on an infinite chessboard reachable by the (3,4)-leaper in <= n moves.

Original entry on oeis.org

1, 9, 41, 129, 321, 681, 1289, 2121, 3081, 4121, 5233, 6445, 7777, 9233, 10813, 12517, 14345, 16297, 18373, 20573, 22897, 25345, 27917, 30613, 33433, 36377, 39445, 42637, 45953, 49393, 52957, 56645, 60457, 64393, 68453, 72637, 76945, 81377, 85933, 90613, 95417

Offset: 0

R. J. Mathar, Jan 05 2018

nonn

Cf. A018836 (1,2)-leaper or (1,3)-leaper, A297740 (2,3)-leaper.

Conjecture: a(n) = 62*n^2 + 30*n - 55 for n >= 10.
Conjectures from Colin Barker, Jan 06 2018: (Start)
G.f.: (1 + 6*x + 17*x^2 + 32*x^3 + 48*x^4 + 64*x^5 + 80*x^6 - 24*x^7 - 96*x^8 - 48*x^9 - 8*x^10 + 28*x^11 + 20*x^12 + 4*x^13) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>13.
(End)

A118312 Number of squares on infinite chessboard that a knight can reach in n moves from a fixed square.

Original entry on oeis.org

1, 8, 33, 76, 129, 196, 277, 372, 481, 604, 741, 892, 1057, 1236, 1429, 1636, 1857, 2092, 2341, 2604, 2881, 3172, 3477, 3796, 4129, 4476, 4837, 5212, 5601, 6004, 6421, 6852, 7297, 7756, 8229, 8716, 9217, 9732, 10261, 10804, 11361, 11932, 12517, 13116, 13729, 14356, 14997, 15652

Offset: 0

Anton Chupin (chupin(AT)icmm.ru), May 14 2006

Related to A018842: a(n) = A018842(n) + A018842(n-2) + A018842(n-4) + ... .

			a(2)=33 because knight in 2 moves from square (0,0) can reach one of the following squares: {{0,0}, {-4,-2}, {-4,0}, {-4,2}, {-3,-3}, {-3,-1}, {-3,1}, {-3,3}, {-2,-4}, {-2,0}, {-2,4}, {-1,-3}, {-1,-1}, {-1,1}, {-1,3}, {0,-4}, {0,-2}, {0,2}, {0,4}, {1,-3}, {1,-1}, {1,1}, {1,3}, {2,-4}, {2,0}, {2,4}, {3,-3}, {3,-1}, {3,1}, {3,3}, {4,-2}, {4,0}, {4,2}}.

M. Petkovic, Mathematics and Chess, Dover Publications (2003), Problem 3.11.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mordechai Katzman, Knight's moves on an infinite board
Mordechai Katzman, Counting monomials, arXiv:math/0504113 [math.AC], 2005.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005892, A018842 (squares in EXACTLY n moves), A018836 (squares in <=n moves).

I:=[1, 8, 33, 76, 129, 196, 277]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 09 2012

Table[ -3 + 4*n + 7*n^2 + 4*Sign[(n - 2)(n - 1)], {n, 0, 100}]
CoefficientList[Series[(1+5*x+12*x^2-8*x^4+4*x^5)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 09 2012 *)
Join[{1,8,33},LinearRecurrence[{3,-3,1},{76,129,196},50]] (* Harvey P. Dale, Dec 05 2014 *)

a(n)=7*n^2 + 4*n - 3 + 4*sign((n-2)*(n-1)) \\ Charles R Greathouse IV, Feb 09 2017

a(n) = -3 + 4*n + 7*n^2 + 4*sign((n-2)*(n-1)).
G.f.: (1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 09 2012
For n >= 3, a(n) = A005892(n).
E.g.f.: exp(x)*(1 + 11*x + 7*x^2) - 2*x*(x + 2). - Stefano Spezia, Jul 27 2022

Link updated by Tristan Miller, Jun 13 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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A297740 The number of distinct positions on an infinite chessboard reachable by the (2,3)-leaper in <= n moves.

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A018842 Number of squares on infinite chessboard at n knight's moves from center.

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

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A297741 The number of distinct positions on an infinite chessboard reachable by the (3,4)-leaper in <= n moves.

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A118312 Number of squares on infinite chessboard that a knight can reach in n moves from a fixed square.

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