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A254129
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Number of 2n-move closed knight paths on an unbounded chessboard from a given square to the same square.
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9
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1, 8, 168, 5840, 261800, 13180608, 702273264, 38641656768, 2171652448680, 123938999632448, 7158206751686848, 417418594698260064, 24535017440445455216, 1451786144317963971200, 86396682439552099487040, 5166936574734171792925440, 310340697572034456203934120
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OFFSET
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0,2
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COMMENTS
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Every move changes the color of the square the knight is on, so there is no returning path of odd length.
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LINKS
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FORMULA
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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)
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
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EXAMPLE
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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.
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MAPLE
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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:
# 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):
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MATHEMATICA
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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];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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