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 1-3 of 3 results.

A254129 Number of 2n-move closed knight paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5840, 261800, 13180608, 702273264, 38641656768, 2171652448680, 123938999632448, 7158206751686848, 417418594698260064, 24535017440445455216, 1451786144317963971200, 86396682439552099487040, 5166936574734171792925440, 310340697572034456203934120
Offset: 0

Views

Author

David A. Corneth, Jan 25 2015

Keywords

Comments

Every move changes the color of the square the knight is on, so there is no returning path of odd length.

Examples

			a(1) = 8. For illustration, let's assume we're on a usual 8 X 8 chessboard, with the knight initially at D4. Then there are 8 paths bringing it back to D4 in 2 moves:
D4-E6-D4; D4-F5-D4; D4-F3-D4; D4-E2-D4; D4-C2-D4; D4-B3-D4; D4-B5-D4; D4-C6-D4.

Links

Crossrefs

Cf. A025600, A094061, A253974, A254459, A307347.

Programs

  • Maple
    G:= cos(x+2*y)+cos(x-2*y)+cos(2*x+y)+cos(2*x-y):
F:= 1: a[0]:= 1:
for n from 1 to 20 do
  F:= combine(F*G^2,trig);
  a[n]:= 4^n*remove(has,F,cos);
od:
seq(a[n],n=0..20);  # Robert Israel, Jan 26 2015
# second Maple program:
b:= proc(n, x, y) option remember; `if`(max(x, y)>2*n or x+y>3*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[1, 2],
      [2, 1], [-1, 2], [-2, 1], [1, -2], [2, -1], [-1, -2], [-2, -1]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 29 2015
  • Mathematica
    b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 2n || x+y > 3n, 0, If[n == 0, 1, Sum[b[n-1, Abs[x+l[[1]]], Abs[y+l[[2]]]], {l, {{1, 2}, {2, 1}, {-1, 2}, {-2, 1}, {1, -2}, {2, -1}, {-1, -2}, {-2, -1}}}]]];
a[n_] :=  b[2n, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 30 2019, after Alois P. Heinz *)
  • PARI
    a(n)={my(l=listcreate(),v=vector(2*n+1));m=matrix(1,1);m[1,1]=1;listput(l,m);v[1]=1;for(i=2,2*n+1, m=matrix(4*i-3,4*i-3);for(j=1,#l[i-1],for(k=1,#l[i-1],m[j+2-2,k+2-1]+=l[i-1][j,k];m[j+2-2,k+2+1]+=l[i-1][j,k];m[j+2-1,k+2-2]+=l[i-1][j,k];m[j+2-1,k+2+2]+=l[i-1][j,k];m[j+2+1,k+2-2]+=l[i-1][j,k];m[j+2+1,k+2+2]+=l[i-1][j,k];m[j+2+2,k+2-1]+=l[i-1][j,k];m[j+2+2,k+2+1]+=l[i-1][j,k]));v[i]=m[2*i-1,2*i-1];listput(l,m););listput(l,v);v[#v]} \\ David A. Corneth, Jan 26 2015
  • PARI
    {a(n) = polcoef(polcoef((x^2*y+x*y^2+y^2/x+y/x^2+1/(x^2*y)+1/(x*y^2)+x/y^2+x^2/y)^(2*n), 0), 0)} \\ Seiichi Manyama, Nov 02 2019

Formula

From Robert Israel, Jan 26 2015: (Start)
a(n) = 4^n * (2*Pi)^(-2) * int_0^(2*Pi) int_0^(2*Pi) ds dt (cos(s+2*t)+cos(s-2*t)+cos(2*s+t)+cos(2*s-t))^(2*n).
G.f.: (2*Pi)^(-2) * int_0^(2*Pi) int_0^(2*Pi) ds dt 1/(1 - 4*x*(cos(s+2*t)+cos(s-2*t)+cos(2*s+t)+cos(2*s-t))^2).
(End)
a(n) ~ 64^n / (5*Pi*n). - Vaclav Kotesovec, Jan 28 2015
Recurrence (conjectured): 3*n^2*(3*n-2)*(3*n-1)*(4*n-3)*(4*n-1)*(58625*n^6 - 574525*n^5 + 2317575*n^4 - 4929815*n^3 + 5836090*n^2 - 3647730*n + 940788)*a(n) = 4*(563444875*n^12 - 7212094400*n^11 + 40894216825*n^10 - 135653664390*n^9 + 292742658975*n^8 - 432166599360*n^7 + 446527351283*n^6 - 324481592710*n^5 + 164046706898*n^4 - 56035458036*n^3 + 12203976528*n^2 - 1507156200*n + 78246000)*a(n-1) - 64*(n-1)*(2*n-3)^2*(167726125*n^9 - 1643716025*n^8 + 6735239425*n^7 - 15048594215*n^6 + 20072439970*n^5 - 16473493280*n^4 + 8273936628*n^3 - 2437948332*n^2 + 377982648*n - 22556880)*a(n-2) + 165888*(n-2)*(n-1)*(2*n - 5)^2*(2*n - 3)^2*(58625*n^6 - 222775*n^5 + 324325*n^4 - 232265*n^3 + 86220*n^2 - 15570*n + 1008)*a(n-3). - Vaclav Kotesovec, Jan 30 2015
a(n) = the constant term in the expansion of (x^2*y + x*y^2 + y^2/x + y/x^2 + 1/(x^2*y) + 1/(y^2*x) + x/y^2 + x^2/y)^(2*n). - Peter Bala, Feb 14 2017
The conjectured recurrence of Vaclav Kotesovec is true. Running the input file inSMAZ6 (see Links) on the Maple program SMAZ gives the recurrence followed by the certificate shown in the output file oSMAZ6. - Doron Zeilberger, Mar 29 2019

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

Views

Author

Vaclav Kotesovec, Jan 31 2015

Keywords

Comments

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.

Links

Crossrefs

Cf. A094061, A254129, A254459, A307347.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Views

Author

Vaclav Kotesovec, Jan 30 2015

Keywords

Comments

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.

Links

Crossrefs

Cf. A094061, A253974, A254129, A307347.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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
Showing 1-3 of 3 results.

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