A098498 -id:A098498 - cp's OEIS frontend

A018836 Number of squares on infinite chessboard at <= n knight's moves from a fixed square.

1, 9, 41, 109, 205, 325, 473, 649, 853, 1085, 1345, 1633, 1949, 2293, 2665, 3065, 3493, 3949, 4433, 4945, 5485, 6053, 6649, 7273, 7925, 8605, 9313, 10049, 10813, 11605, 12425, 13273, 14149, 15053, 15985, 16945, 17933, 18949, 19993, 21065, 22165

Offset: 0

N. J. A. Sloane, Marc LeBrun

Apparently also the number of distinct squares reachable by the (1,3)-leaper in at most n moves. - R. J. Mathar, Jan 05 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Erich Friedman, Illustration of initial terms
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 (3,-3,1).

Partial sums of A018842. Cf. A098498 (half-infinite board), A001844 (1,1)-leaper, A297740 (2,3)-leaper, A297741 (3,4)-leaper.

(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^3; seq(coeff(series(%, x, n+1), x, n), n=0..50);

Table[1-6 n+14 n^2+4 Sign[n(n-1)(n-3)], {n, 0, 50}] (* Zak Seidov *)
Join[{1,9,41,109},LinearRecurrence[{3,-3,1},{205,325,473},50]] (* Harvey P. Dale, Aug 16 2011 *)
CoefficientList[Series[(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1 + x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 26 2012 *)

G.f.: (1+5*x+12*x^2-8*x^4+4*x^5)*(1+x)/(1-x)^3;

a(n) = 1-6*n+14*n^2+4*sign(n*(n-1)*(n-3)). - Zak Seidov, Mar 01 2005

A098502 a(n) = 16*n - 4.

Original entry on oeis.org

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Ralf Stephan, Sep 15 2004

For n > 3, the number of squares on the infinite 4-column chessboard at <= n knight moves from any fixed start point.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences which agree for a long time but are different.

Cf. A004767, A008590, A017113, A017137, A017629, A098498, A158953.

[16*n - 4: n in [1..60]]; // Vincenzo Librandi, Jul 24 2011

Range[12, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

a(n)=16*n-4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: 4*x*(3+x)/(1-x)^2. - Colin Barker, Jan 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3 - 2*sqrt(2)))/(16*sqrt(2)). - Amiram Eldar, Sep 01 2024
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = 4*A004767(n-1) = 2*A017137(n-1) = A017113(2*n-1). (End)

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

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

A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.

Original entry on oeis.org

1, 2, 10, 22, 37, 54, 76, 100, 129, 160, 196, 234, 277, 322, 372, 424, 481, 540, 604, 670, 741, 814, 892, 972, 1057, 1144, 1236, 1330, 1429, 1530, 1636, 1744, 1857, 1972, 2092, 2214, 2341, 2470, 2604, 2740, 2881, 3024, 3172, 3322, 3477, 3634, 3796, 3960

Offset: 0

Bodo Zinser, Nov 18 2001

The first conjecture is true: Partial sums of A047356 = b(n) = (14*(n*(n+1)) + 2*n + 5 + 3*(-1)^n )/8, since A047356(n)=(14*n+1+3*(-1)^n)/4. And b(n) has g.f. (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). The difference a(n) - b(n) = 0,2,2,0,0,0,0,0,0..., which has g.f. 2*x^2 + 2*x. Then (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1) + 2*x^2 + 2*x = (-2*x^6 + 2*x^5 + 4*x^4 - 4*x^3 + 2*x^2 + 4*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). - Vim Wenders, Apr 16 2008

Cf. A098498.

Conjectures: G.f.: [1+6x^2+4x^3-4x^4-2x^5+2x^6]/[(1+x)*(1-x)^3]. For n>3, partial sums of A047356. - Ralf Stephan, Mar 06 2004

The second conjecture "For n>3, partial sums of A047356" is also true. From the last possible move, we can either move back to the second last possible move or to b(n)=A047883(n) new squares. So a(n) = a(n-2)+b(n). For n>6, b(n)=7(n-1)+4=A017029(n-1). But a number of the form 7n+4 is naturally the sum of two consecutive terms in A047356 (4=1+3,11=3+8,18=8+10, ...). The conjecture follows. - Vim Wenders, Apr 12 2008

More terms from Don Reble, Nov 28 2001

cp's OEIS Frontend

A018836 Number of squares on infinite chessboard at <= n knight's moves from a fixed square.

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A098502 a(n) = 16*n - 4.

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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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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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A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.

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