A005635 -id:A005635 - cp's OEIS frontend

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

1, 1, 2, 7, 23, 115, 694, 5282, 46066, 456454, 4999004, 59916028, 778525516, 10897964660, 163461964024, 2615361578344, 44460982752488, 800296985768776, 15205638776753680, 304112757426239984, 6386367801916347184

Offset: 1

N. J. A. Sloane

			For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
Eric Weisstein's World of Mathematics, Rook Number.
Eric Weisstein's World of Mathematics, Rooks Problem.

Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903
P:=n->n!; # Gives A000142
G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)
rho:=n->R(n)/2; # Gives A000407, aerated
beta:=n->B(n)/2; # Gives A000902, doubled up
delta:=n->(D(n)-B(n))/2; # Gives A000900
unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up
alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

c[n_] := Floor[n/2]! 2^Floor[n/2];
r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
Array[a, 21] (* Jean-François Alcover, Apr 06 2011, after Matthias Engelhardt, further improved by Robert G. Wilson v *)

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e., factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions] and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal]. - Matthias Engelhardt, Apr 05 2000

For asymptotics see the Robinson paper.

More terms from David W. Wilson, Jul 13 2003

A122749 Number of arrangements of n non-attacking bishops on an n X n board such that every square of the board is controlled by at least one bishop.

Original entry on oeis.org

4, 2, 16, 44, 256, 768, 5184, 25344, 186624, 996480, 8294400, 57888000, 530841600, 4006195200, 40642560000, 367408742400, 4064256000000, 39358255104000, 474054819840000, 5254107586560000, 68263894056960000, 804207665479680000, 11242684107325440000

Offset: 2

N. J. A. Sloane, Sep 25 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 2..200
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [E_n, n >= 2.]

Cf. A005635, A182333.

E:=proc(n) local k; if n mod 2 = 0 then k := n/2; if k mod 2 = 0 then RETURN( (k!*(k+2)/2)^2 ); else RETURN( ((k-1)!*(k+1)^2/2)^2 ); fi; else k := (n-1)/2; if k mod 2 = 0 then RETURN( ((k!)^2/12)*(3*k^3+16*k^2+18*k+8) ); else RETURN( ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3) ); fi; fi; end;

Table[If[n==1,1,1/768*(2*(3*n^3+23*n^2+17*n+21)*(((n-1)/2)!)^2*(1-(-1)^n+2*Sin[(Pi*n)/2])-2*(3*n^3+17*n^2-47*n+3)*((n-3)/2)!*((n+1)/2)!*((-1)^n+2*Sin[(Pi*n)/2]-1)+3*(n+2)^4*((n/2-1)!)^2*((-1)^n-2*Cos[(Pi*n)/2]+1)+12*(n+4)^2*((n/2)!)^2*((-1)^n+2*Cos[(Pi*n)/2]+1))],{n,2,25}] (* Vaclav Kotesovec, Apr 26 2012 *)
a[n_] := Module[{k}, If[Mod[n, 2]==0, k = n/2; If[Mod[k, 2]==0, (k!*(k+2) /2)^2, ((k-1)!*(k+1)^2/2)^2], k = (n-1)/2; If[Mod[k, 2]==0, ((k!)^2/12)* (3*k^3+16*k^2+18*k+8), ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3)]]];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Jul 23 2022, after Maple code *)

From Andy Huchala, Mar 22 2024 (based on Mathematica code): (Start)
a(4*n)   = (n+1)^2*((2*n)!)^2.
a(4*n+1) = (1/3)*(n+2)*(1+4*n+6*n^2)*((2*n)!)^2.
a(4*n+2) = 4*(n+1)^4*((2*n)!)^2.
a(4*n+3) = (1/3)*(3+17*n+22*n^2+6*n^3)*(2*n)!*(2*n+2)!.  (End)

New name from Vaclav Kotesovec, Apr 26 2012

A182333 Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop.

Original entry on oeis.org

1, 4, 6, 25, 104, 484, 2136, 11664, 71136, 451584, 3006720, 21902400, 176774400, 1456185600, 12758860800, 117456998400, 1181072793600, 12023694950400, 130072449024000, 1451792885760000, 17487355576320000, 212389727477760000, 2729844680048640000

Offset: 1

Vaclav Kotesovec, Apr 25 2012

nonn

Number of minimum dominating sets in the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, vol.1, 1987, p.11 and p.83-88.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Dominating Set

Cf. A005635, A122749, A323500, A323501.

Table[If[n==1,1,((2*Floor[n/4])!)^2/128*(n^5+3*n^4+n^3+35*n^2+38*n+2-(n^5-n^4-7*n^3-n^2-10*n-30)*(-1)^n-4*(n^3+2*n^2+n-4)*n*Cos[Pi*n/2]-2*(n^5+n^4-11*n^3-7*n^2-2*n+2)*Sin[Pi*n/2])],{n,1,25}]

a(n)={if(n==1, 1, (n\4*2)!^2*if(n%4<2, if(n%2==0, (n+1)^2, (n^3 + 3*n^2 + 2*n - 2)/2), if(n%2==0, (n^2+n+2)^2/4, (n+1)*(n-1)*(n^3 + n^2 - 6*n + 6)/8))/4)} \\ Andrew Howroyd, Sep 09 2019

a(n) = (((2*floor(n/4))!)^2/128)*(n^5 + 3*n^4 + n^3 + 35*n^2 + 38*n + 2 - (n^5 - n^4 - 7*n^3 - n^2 - 10*n - 30)*(-1)^n -4*(n^3 + 2*n^2 + n - 4)*n*cos(Pi*n/2) - 2*(n^5 + n^4 - 11*n^3 - 7*n^2 - 2*n + 2)*sin(Pi*n/2)), for n > 1.

a(n) = A323500(n) * A323501(n) for n > 1. - Andrew Howroyd, Sep 08 2019

A123072 Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).

Original entry on oeis.org

1, 2, 72, 7200, 1411200, 457228800, 221298739200, 149597947699200, 134638152929280000, 155641704786247680000, 224746621711341649920000, 396453040698806670458880000, 838894634118674914690990080000, 2097236585296687286727475200000000, 6115541882725140128097317683200000000

Offset: 0

N. J. A. Sloane, Sep 28 2006

nonn

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence zeta(2k+1).]

Cf. A173331. [Reinhard Zumkeller, Feb 16 2010]

Cf. A001700, A010050, A156992, A187535.

Maple
```
For Maple program see A005635.
```

Table[(((2 n)!/n!)^2)/2, {n, 1, 20}] (* Benedict W. J. Irwin, Jun 05 2016 *)
Table[SeriesCoefficient[Series[1/2 + EllipticK[16 x]/Pi, {x, 0, 20}],n] n! n!, {n, 1, 20}] (* Benedict W. J. Irwin, Jun 05 2016 *)

From_Reinhard Zumkeller_, Feb 16 2010: (Start)
a(n) = ceiling((((2*n)! / n!)^2) / 2).
a(n) = A001700(n-1) * A010050(n). (End)
From Benedict W. J. Irwin, Jun 05 2016: (Start)
G.f. for a(n)/(n!)^2 : 1/2 + EllipticK(16*x)/Pi, which is the E.g.f for A187535.
G.f. for a(n)/(n!)^3 : 2F2(1/2, 1/2; 1, 1; 16z)/2.
a(n) = n!*A187535(n) = binomial(2*n-1, n-1)*(2*n)!.
(End)
a(n) = A156992(2n,n). - Alois P. Heinz, Apr 30 2017
a(n) ~ asy(2*n-1) where asy(n) = (2*n/e)^n*(18*n + 6 + 1/n)/9. - Peter Luschny, Dec 05 2019
Sum_{n>=0} 1/a(n) = 1 + StruveL(0, 1/2)*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Dec 04 2022

a(0)=1 prepended by Alois P. Heinz, Apr 30 2017

A123071 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

Original entry on oeis.org

1, 2, 4, 12, 36, 120, 400, 1520, 5776, 23712, 97344, 431808, 1915456, 9012608, 42406144, 210988800, 1049760000, 5475340800, 28558296064, 155672726528, 848579961856, 4810614454272, 27271456395264, 160376430784512, 943132599095296, 5735299537018880

Offset: 0

N. J. A. Sloane, Sep 28 2006

nonn

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence S(2k+1) eq(24) p. 210.]
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Cf. A000898, A005635, A135401 (B(n)), A202828.

For Maple program see A005635.
# alternative
# this is A000898, replicated as 1,1,2,2,6,6,20,20,76,76,...
B := proc(n)
    if n=0 or n= -2 then
        1 ;
    elif type (n,'odd') then
        procname(n-1) ;
    else
        2*procname(n-2)+(n-2)*procname(n-4) ;
    end if;
end proc:
A123071 := proc(n)
    B(n)*B(n+1) ;
end proc:
seq(A123071(n),n=0..20) ; # R. J. Mathar, Apr 02 2017

B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n-1], True, 2*B[n-2] + (n-2)*B[n-4]];
a[n_] := B[n]*B[n+1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 23 2022, after R. J. Mathar *)

Conjecture: 2*a(n) +a(n-1) -2*n*a(n-2) +(-n-10)*a(n-3) -2*(n-2)*(n+2)*a(n-4) +(-n^2-2*n+23)*a(n-5) +2*(n-5)*(n^2-7*n+11)*a(n-6) +(n-6)*(n-5)^2*a(n-7)=0. - R. J. Mathar, Apr 02 2017

A005631 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

Original entry on oeis.org

1, 2, 6, 18, 60, 200, 760, 2888, 11856, 48672, 215904, 957728, 4506304, 21203072, 105494400, 524880000, 2737670400, 14279148032, 77836363264, 424289980928, 2405307227136, 13635728197632, 80188215392256, 471566299547648, 2867649768509440, 17438513317683200

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence psi(2k+1).]
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Equals A123071(n)/2, n >= 1.

Cf. A005635.

Maple
```
For Maple program see A005635.
```

B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n - 1], True, 2*B[n - 2] + (n - 2)*B[n - 4]];
a[n_] := B[n + 1]*B[n + 2]/2;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 23 2022, after Maple code for A123071 *)

More terms from N. J. A. Sloane, Sep 28 2006

A005632 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

Original entry on oeis.org

0, 0, 5, 22, 258, 1628, 18052, 145976, 1837272, 18407664, 265312848, 3184567136, 52020223648, 728304073664, 13317701313600, 213083801827200, 4314950946864000, 77669134543011584, 1725980887361498368, 34519618313219995136, 835374767116711506432, 18378244896208168541184

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence mu(2k+1).]
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Maple
```
For Maple program see A005635.
```

c[n_] := Module[{k}, If[Mod[n, 2]==0, Return[0]]; k = (n-1)/2; If[Mod[k, 2] == 0, Return[k*2^(k-1)*((k/2)!)^2], Return[2^k*(((k+1)/2)!)^2]]];
d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];
B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n-1], True, 2*B[n-2] + (n - 2)*B[n - 4]];
S[n_] := S[n] = Module[{k}, If[Mod[n, 2]==0, 0, k = (n-1)/2; B[k]*B[k+1]]];
Q[n_] := Module[{m}, If[Mod[n, 8] != 1, Return[0]]; m = (n-1)/8; ((2*m)!)^2 /(m!)^2];
a[n_] := (c[2n+1] - S[2n+1] - Q[2n+1])/4;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jul 23 2022, after Maple program in A005635 *)

More terms from N. J. A. Sloane, Sep 28 2006

A005633 Bishops on an n X n board (see Robinson paper for details).

Original entry on oeis.org

0, 1, 0, 2, 2, 8, 14, 36, 112, 216, 928, 1440, 8616, 11520, 87864, 100800, 997952, 1008000, 12427904, 10886400, 169435936, 130636800, 2501216992, 1676505600, 39837528576, 23471078400, 679494214656, 348713164800, 12370158205568, 5579410636800, 239109033342848

Offset: 1

N. J. A. Sloane

nonn

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence mu(n).]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Maple
```
For Maple program see A005635.
```

d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];
M[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[k!*(k + 2)/2], Return[(k - 1)!*(k + 1)^2/2]], k = (n - 1)/2; Return[d[k]*d[k + 1]]]];
B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n - 1], True, 2*B[n - 2] + (n - 2)*B[n - 4]];
S[n_] := S[n] = Module[{k}, If[Mod[n, 2]==0, 0, k = (n-1)/2; B[k]*B[k+1]]];
a[n_] := (M[n] - S[n])/2;
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 23 2022, after Maple program in A005635 *)

More terms from N. J. A. Sloane, Sep 28 2006

A005634 Bishops on an n X n board (see Robinson paper for details).

Original entry on oeis.org

0, 0, 1, 4, 28, 85, 630, 3096, 23220, 123952, 1036080, 7230828, 66349440, 500721252, 5080269600, 45925520096, 508031496000, 4919774752448, 59256847036800, 656763354386032, 8532986691801600, 100525956801641104, 1405335512577427200, 18431883446961030912

Offset: 2

N. J. A. Sloane

nonn

The problem of the bishops is to determine the number of inequivalent arrangements of n bishops on an n X n chessboard such that no bishop threatens another and every unoccupied square is threatened by some bishop. Two arrangements are considered equivalent if they are isomorphic by way of one of the eight symmetries of the chessboard. - Jean-François Alcover, Jul 24 2022 (after Robinson's paper).

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence epsilon(n) page 212.]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Maple
```
For Maple program see A005635.
```

e[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[(k!*(k + 2)/2)^2], Return[((k - 1)!*(k + 1)^2/2)^2]], k = (n - 1)/2; If[Mod[k, 2] == 0, Return[((k!)^2/12)*(3*k^3 + 16*k^2 + 18*k + 8)], Return[((k - 1)!*(k + 1)!/12)*(3*k^3 + 13*k^2 - k - 3)]]]];
c[n_] := Module[{k}, If[Mod[n, 2]==0, Return[0]]; k = (n-1)/2; If[Mod[k, 2] == 0, Return[k*2^(k-1)*((k/2)!)^2], Return[2^k*(((k+1)/2)!)^2]]];
d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];
B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n - 1], True, 2*B[n - 2] + (n - 2)*B[n - 4]];
S[n_] := S[n] = Module[{k}, If[Mod[n, 2]==0, 0, k = (n-1)/2; B[k]*B[k+1]]];
M[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[k!*(k + 2)/2], Return[(k - 1)!*(k + 1)^2/2]], k = (n - 1)/2; Return[d[k]*d[k + 1]]]];
a[n_] := e[n]/8 - c[n]/8 + S[n]/4 - M[n]/4;
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jul 23 2022, after Maple program in A005635 *)

More terms from N. J. A. Sloane, Sep 28 2006

cp's OEIS Frontend

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

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A122749 Number of arrangements of n non-attacking bishops on an n X n board such that every square of the board is controlled by at least one bishop.

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A182333 Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop.

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A123072 Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).

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A123071 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

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A005631 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

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Maple

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A005632 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

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Maple

Mathematica

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A005633 Bishops on an n X n board (see Robinson paper for details).