A254129 -id:A254129 - cp's OEIS frontend

A094061 Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard.

1, 0, 8, 24, 216, 1200, 8840, 58800, 423640, 3000480, 21824208, 158964960, 1171230984, 8668531872, 64574844048, 483114856224, 3630440899800, 27379154692032, 207172490054816, 1572194644061184, 11962847247681616, 91242602561647680, 697438669619791008

Offset: 0

Matthijs Coster, Apr 29 2004

The chessboard here is the full four-quadrant board Z X Z.
This is an analog of A054474 for walks on a square grid where the steps can be made diagonally as well.
a(n) is the constant term in the expansion of ((x + 1/x) * (y + 1/y) + x^2 + 1/x^2 + y^2 + 1/y^2)^n. - Seiichi Manyama, Nov 03 2019

D. Joyner, "Adventures in Group Theory: Rubik's Cube, Merlin's Machine and Other Mathematical Toys", Johns Hopkins University Press, 2002, pp. 79

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Peter Bala, A note on A094061

Row 2 of A327751.

Cf. A002426, A098070, A126869, A253974, A254129, A254459, A329024.

a:=array(0..30):a[0]:=1:a[1]:=0:a[2]:=8:a[3]:=24:for n from 3 to 29 do a[n+1]:= (n*(5*n+1)*a[n]+2*(15*n^2+6*n-5)*a[n-1]-8*(5*n^2-23*n+21)*a[n-2]-64*(n-2)^2*a[n-3])/(n+1)^2: print(n+1,a[n+1]) od:
# second Maple program
a:= proc(n) option remember; `if`(n<3, (n-1)*(9*n-2)/2,
      ((n-1)*(3*n-1)*(3*n-4) *a(n-1)
      +(108*n^3-396*n^2+452*n-152) *a(n-2)
      +32*(3*n-2)*(n-2)^2 *a(n-3))/ (n^2*(3*n-5)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

a[n_]:=Module[{f=(x+x^-1+y+y^-1+x y+x^-1y+x^-1y^-1+x y^-1)^n,s}, s=Series[f,{x,0,0},{y,0,0}]; SeriesCoefficient[s,{0,0}]]; Table[a[n], {n,1,22}] (* Armin Vollmer (Armin.Vollmer(AT)kabelleipzig.de), May 01 2006 *)
CoefficientList[Series[1/(1+4*x)*LegendreP[-1/2,1-32*x*(1+x)/(1+4*x)^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 16 2013 *)

a[0]:1$
a[1]:0$
a[2]:8$
a[3]:24$
a[n]:=((n-1)*(3*n-1)*(3*n-4) *a[n-1]
      +(108*n^3-396*n^2+452*n-152) *a[n-2]
      +32*(3*n-2)*(n-2)^2 *a[n-3])/ (n^2*(3*n-5))$
A094061(n):=a[n]$
makelist(A094061(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

{a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^2)} \\ Seiichi Manyama, Oct 29 2019

{a(n) = polcoef(polcoef(((x+1/x)*(y+1/y)+x^2+1/x^2+y^2+1/y^2)^n, 0), 0)} \\ Seiichi Manyama, Nov 03 2019

D-finite with recurrence (n+1)^2*a(n+1) = n*(5*n+1)*a(n) + 2*(15*n^2+6*n-5)*a(n-1) - 8*(5*n^2-23*n+21) *a(n-2) - 64*(n-2)^2*a(n-3).
G.f.: (2/(Pi*(1+4*x))) * EllipticK(4*sqrt(x*(1+x))/(1+4*x)) = 1/(1+4*x) * hypergeom([1/2,1/2], [1], 16*(x*(1+x))/(1+4*x)^2). - Sergey Perepechko, Jan 15 2011
a(n) ~ 2^(3*n+1)/(3*Pi*n). - Vaclav Kotesovec, Aug 16 2013
a(n) = (1/Pi^2) * Integral_{y = 0..Pi} Integral_{x = 0..Pi} (2*cos(x) + 2*cos(y) + 4*cos(x)*cos(y))^n dx dy. - Peter Bala, Feb 14 2017
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^2. - Seiichi Manyama, Oct 29 2019
From Peter Bala, Feb 08 2022: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the stronger congruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and positive integers n and k. (End)
a(n) = Sum_{j = 0..n} Sum_{k = 0..j} binomial(2*j,j)^2*binomial(j,k)* binomial(n+j-k,2*j)*(-4)^(n-j-k). - Peter Bala, Mar 19 2022

More terms from and entry improved by Sergey Perepechko, Sep 06 2004

A253974 Number of 2n-move closed giraffe paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 7964208, 362370624, 17532536736, 889716433320, 46887220540160, 2546408317827088, 141659449976239104, 8033749056463329472, 462687411167492828000, 26980019699392099317600, 1589091557661690119997120, 94361786346423775855372200

Offset: 0

Vaclav Kotesovec, Jan 31 2015

nonn

Giraffe is a (fairy chess) leaper [1,4].

Conjecture: Number of 2n-move closed paths of leaper [r,s] on an unbounded chessboard, where 0 < r < s and gcd(r,s)=1, is asymptotic to 2^(6*n+1) / ((r^2+s^2)*Pi*n) if r+s is even, and 2^(6*n) / ((r^2+s^2)*Pi*n) if r+s is odd.

Vaclav Kotesovec, Table of n, a(n) for n = 0..552
Vaclav Kotesovec, Conjectured recurrence (of order 8)
Vaclav Kotesovec, Examples of closed giraffe paths
Wikipedia, Fairy chess piece

Cf. A094061, A254129, A254459, A307347.

b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>5*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 1],
      [1, 4], [-4, 1], [-1, 4], [4, -1], [1, -4], [-4, -1], [-1, -4]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25); # after Alois P. Heinz
# second Maple program:
poly:=expand((x*y^4+x^4*y+y^4/x+y/x^4+x/y^4+x^4/y+1/(x*y^4)+1/(x^4*y))^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do: # Vaclav Kotesovec, Apr 03 2019

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 4n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{4, 1}, {1, 4}, {-4, 1}, {-1, 4}, {4, -1}, {1, -4}, {-4, -1}, {-1, -4}}}]]];
a[n_] := b[2n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Maple *)

a(n) ~ 64^n / (17*Pi*n).

a(n) = the constant term in the expansion of (x*y^4 + x^4*y + 1/x*y^4 + 1/x^4*y + x/y^4 + x^4/y + 1/x/y^4 + 1/x^4/y)^(2*n). - Vaclav Kotesovec, Apr 01 2019

A254459 Number of 2n-move closed zebra paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 8039808, 373369920, 18576523680, 972362837160, 52832252432960, 2950644716576128, 168192125309339040, 9735527029198105408, 570163460613978204800, 33697054064651581144800, 2005939326990647575285920, 120109818840839172931095720

Offset: 0

Vaclav Kotesovec, Jan 30 2015

nonn

Zebra is a (fairy chess) leaper [2,3].

Conjecture: Number of 2n-move closed paths of leaper [r,s] on an unbounded chessboard, where 0 < r < s and gcd(r,s)=1, is asymptotic to 2^(6*n+1) / ((r^2+s^2)*Pi*n) if r+s is even, and 2^(6*n) / ((r^2+s^2)*Pi*n) if r+s is odd.

Vaclav Kotesovec, Table of n, a(n) for n = 0..542
Vaclav Kotesovec, Examples of closed zebra paths
Vaclav Kotesovec, Conjectured recurrence (of order 6)
Wikipedia, Fairy chess piece

Cf. A094061, A253974, A254129, A307347.

b:= proc(n, x, y) option remember; `if`(max(x, y)>3*n or x+y>5*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[3, 2],
      [2, 3], [-3, 2], [-2, 3], [3, -2], [2, -3], [-3, -2], [-2, -3]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25); # after Alois P. Heinz
# second Maple program:
poly:=expand((x^2*y^3 + x^3*y^2 + 1/x^2*y^3 + 1/x^3*y^2 + x^2/y^3 + x^3/y^2 + 1/x^2/y^3 + 1/x^3/y^2)^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do: # Vaclav Kotesovec, Apr 03 2019

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 3n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{3, 2}, {2, 3}, {-3, 2}, {-2, 3}, {3, -2}, {2, -3}, {-3, -2}, {-2, -3}}}]]];
a[n_] := b[2n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Maple *)

a(n) ~ 64^n / (13*Pi*n).

a(n) = the constant term in the expansion of (x^2*y^3 + x^3*y^2 + 1/x^2*y^3 + 1/x^3*y^2 + x^2/y^3 + x^3/y^2 + 1/x^2/y^3 + 1/x^3/y^2)^(2*n). - Vaclav Kotesovec, Apr 01 2019

A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3) - (x + 1/x)(y + 1/y))^(2*n).

Original entry on oeis.org

1, 12, 588, 49440, 5187980, 597027312, 71962945824, 8923789535232, 1128795397492620, 144940851928720848, 18832163401980525168, 2470451402766989534256, 326667449725835512275488, 43485599433527022301377600, 5821983056232777427055717760

Offset: 0

Seiichi Manyama, Nov 02 2019

nonn

Also number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 3).
         *
         |
      *-- --*
      |  |  |
   *-- -- -- --*
   |  |  |  |  |
*-- -- --P-- -- --*
   |  |  |  |  |
   *-- -- -- --*
      |  |  |
      *-- --*
         |
         *
Point P move to any position of * in the next step.

Seiichi Manyama, Table of n, a(n) for n = 0..400 (terms 0..185 from Vaclav Kotesovec)
Vaclav Kotesovec, Recurrence of order 4 (conjectured)

Row n=1 of A329066.

Cf. A002894, A094061, A254129.

{a(n) = polcoef(polcoef(((x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3)-(x+1/x)*(y+1/y))^(2*n), 0), 0)}

{a(n) = polcoef(polcoef((sum(k=0, 3, (x^k+1/x^k)*(y^(3-k)+1/y^(3-k)))-x^3-1/x^3-y^3-1/y^3)^(2*n), 0), 0)}

f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
a(n) = sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoef(f(1)^k*f(0)^(2*n-k), 0)^2)

Conjecture: a(n) ~ 3 * 144^n / (19*Pi*n). - Vaclav Kotesovec, Nov 04 2019

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)

A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 7939008, 357713664, 17010543264, 842994009000, 43192225007360, 2275378947981568, 122724475613935104, 6753785574641857024, 378138077830110886400, 21486835143540141873120, 1236506847203439155401920, 71934214120446285067176360

Offset: 0

Vaclav Kotesovec, Apr 03 2019

nonn

Antelope is a leaper [3,4].

Vaclav Kotesovec, Table of n, a(n) for n = 0..552

Cf. A094061, A253974, A254129, A254459.

b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>7*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 3],
      [3, 4], [-4, 3], [-3, 4], [4, -3], [3, -4], [-4, -3], [-3, -4]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);
# second Maple program:
poly := expand((x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do:

poly = Expand[(x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2]; z = 1; Flatten[{1, Table[z = Expand[z*poly]; z[[1]], {n, 1, 15}]}]

a(n) = the constant term in the expansion of (x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^(2*n).

Conjecture: a(n) ~ 64^n / (25*Pi*n).

A368499 Number of non-congruent simple polygons with 2n sides on the unbounded chessboard such that each side is an edge of the corresponding knight graph.

Original entry on oeis.org

3, 13, 178, 3034, 64877, 1503790, 36930111

Offset: 2

Kaloyan Kapralov, Dec 27 2023

A knight graph is a graph that represents all legal moves of the knight chess piece on a chessboard. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other.
Two polygons in the knight graph are called congruent if one can be transformed into the other by applying one or more of the operations of translation, rotation, and reflection on the chessboard; otherwise, they are non-congruent.
This sequence, in contrast to A366778, considers only simple, i.e., non-self-intersecting polygons.

			For n=2 the a(2)=3 solutions (in standard chess notation) are:
  (a1,c2,d4,b3), (a2,c1,d2,c3), (a2,c1,d3,b3).
For n=3 the a(3)=13 solutions are:
  (a1,b3,a5,c4,e3,c2), (a1,b3,a5,c6,b4,c2), (a1,b3,a5,c6,d4,c2),
  (a1,b3,c5,e6,d4,c2), (a2,b4,c2,d4,e2,c1), (a2,b4,c6,d4,b3,c1),
  (a2,b4,c6,d4,e2,c1), (a2,b4,c6,e5,d3,c1), (a2,b4,d5,c3,e2,c1),
  (a2,b4,d5,f4,d3,c1), (a2,b4,d5,f4,e2,c1), (a2,c1,d3,f4,d5,c3),
  (a2,c1,e2,g3,e4,c3).

Stoyan Kapralov, Valentin Bakoev, and Kaloyan Kapralov, Algorithms for Construction and Enumeration of Closed Knight's Paths, Mathematics and Informatics, 2, (2023), 107-114; arXiv:2304.00565 [math.CO], 2023.
Kaloyan Kapralov, Example for n=3: the 13 non-congruent simple polygons.

Cf. A366778, A001230, A234623, A254129, A356404.

A366778 Number of nonequivalent cycles of length 2n in the (2n+1) X (2n+1) knight graph.

Original entry on oeis.org

3, 25, 480, 12000, 350256, 10780549, 344680960

Offset: 2

Stoyan Kapralov, Dec 15 2023

A knight graph is a graph that represents all legal moves of the knight chess piece on a chessboard. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other.

Two cycles in the knight graph are called equivalent if one can be obtained from another by applying one or more of the operations of translation, rotation, and symmetry on the chessboard; otherwise, they are nonequivalent.

			For n=2 the a(2)=3 solutions (in standard chess notation) are: (a1, c2, d4, b3), (a2, c1, d2, c3), and (a2, c1, d3, b3).
Note that each of these three cycles is non-self-intersecting. For the remaining values of n there are two kind of cycles - self-intersecting and non-self-intersecting. For example, a self-intersecting cycle of length 6 is (a1, c2, b4, a2, c1, b3), while the cycle (a1, c2, e1, f3, d4, b3) is non-self-intersecting.

Stoyan Kapralov, Valentin Bakoev, and Kaloyan Kapralov, Algorithms for Construction and Enumeration of Closed Knight's Paths, Mathematics and Informatics, 2, (2023), 107-114; arXiv:2304.00565 [math.CO], 2023.

Cf. A368499, A001230, A234623, A254129, A356404.

cp's OEIS Frontend

A094061 Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard.

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A253974 Number of 2n-move closed giraffe paths on an unbounded chessboard from a given square to the same square.

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A254459 Number of 2n-move closed zebra paths on an unbounded chessboard from a given square to the same square.

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A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)*(y^3 + y + 1/y + 1/y^3) - (x + 1/x)*(y + 1/y))^(2*n).

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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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A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.

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A368499 Number of non-congruent simple polygons with 2n sides on the unbounded chessboard such that each side is an edge of the corresponding knight graph.

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A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3) - (x + 1/x)(y + 1/y))^(2*n).