A000899 -id:A000899 - cp's OEIS frontend

A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

1, 1, 1, 1, 3, 8, 36, 110, 666, 3250, 23436, 125198, 1037520, 7241272, 66360960, 500827928, 5080370400, 45926666984, 508032504000, 4919789029480, 59256857923200, 656763542278304, 8532986822438400, 100525959568386848, 1405335514253932800, 18431883489984091552

Offset: 0

N. J. A. Sloane

From Vaclav Kotesovec, Apr 26 2012: (Start)
This sequence gives (according to the article by Robinson) the number of inequivalent solutions.
For the total number of all arrangements of n non-attacking bishops such that every square of the board is controlled by at least one bishop, see A122749.
For the total number of all arrangements of n bishops (in any position) such that every square of the board is controlled by at least one bishop, see A182333.
(End)

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

N. J. A. Sloane, Table of n, a(n) for n = 0..250
Jean-François Alcover, Mathematica program.
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)

Cf. A005631, A005632, A005633, A005634, A122749, A123071, A123072, 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; # Gives A122749
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
C:=proc(n) local k; if n mod 2 = 0 then RETURN(0); fi; k:=(n-1)/2; if k mod 2 = 0 then RETURN( k*2^(k-1)*((k/2)!)^2 ); else RETURN( 2^k*(((k+1)/2)!)^2 ); fi; end; # Gives A122693
Q:=proc(n) local m; if n mod 8 <> 1 then RETURN(0); fi; m:=(n-1)/8; ((2*m)!)^2/(m!)^2; end; # Gives A122747
M:=proc(n) local k; if n mod 2 = 0 then k:=n/2; if k mod 2 = 0 then RETURN( k!*(k+2)/2 ); else RETURN( (k-1)!*(k+1)^2/2 ); fi; else k:=(n-1)/2; RETURN(D(k)*D(k+1)); fi; end; # Gives A122748
a:=n-> if n <= 1 then RETURN(1) else E(n)/8 + C(n)/8 + Q(n)/4 + M(n)/4; fi; # Gives A005635
# The following additional Maple programs produce A123071, A005631, A123072, A005633, A005632, A005634
S:=proc(n) local k; if n mod 2 = 0 then RETURN(0) else k:=(n-1)/2; RETURN(B(k)*B(k+1)); fi; end; # Gives A123071
psi:=n->S(n)/2; # Gives A005631
zeta:=n->Q(n)/2; # Gives A123072
mu:=n->(M(n)-S(n))/2; # Gives A005633
chi:=n->(C(n)-S(n)-Q(n))/4; # Gives A005632
eps:=n->E(n)/8-C(n)/8+S(n)/4-M(n)/4; # Gives A005634

Entry revised by N. J. A. Sloane, Sep 25 2006

A000902 Expansion of e.g.f. (1/2)(exp(2x + x^2) + 1).

Original entry on oeis.org

1, 1, 3, 10, 38, 156, 692, 3256, 16200, 84496, 460592, 2611104, 15355232, 93376960, 585989952, 3786534784, 25152768128, 171474649344, 1198143415040, 8569374206464, 62668198184448, 468111364627456, 3568287053001728

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
One more than the number of ordered pairs of minimally intersecting partitions such that p consists of exactly two blocks.
The number of B-orbits in the symmetric space of type DIII, SO_{2n}(C)/GL_n(C) where B is a Borel subgroup of SO_{2n}(C). These are parameterized by "type DIII (n,n)-clans". E.g., for n=2, the a(2)=3 type DIII (2,2)-clans are ++--, --++, and 1212. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

T. D. Noe, Table of n, a(n) for n=0..200
Aram Bingham and Özlem Uğurlu, Sects, rooks, pyramids, partitions and paths for type DIII clans, arXiv:1907.08875 [math.CO], 2019.
Aram Bingham and Özlem Uğurlu, DIII clan combinatorics for the orthogonal Grassmannian, Australasian J. of Combinatorics (2021) Vol. 79, No. 1, 55-86.
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]
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
Édouard Lucas, Théorie des nombres (annotated scans of a few selected pages)
B. Pittel, Where the typical set partitions meet and join, Electron. J. of Combin. 7, R5.
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 1/2 * A000898(n) for n>0.

a000902 n = a000902_list !! n
a000902_list = 1 : 1 : 3 : map (* 2) (zipWith (+)
   (drop 2 a000902_list) (zipWith (*) [2..] $ tail a000902_list))
-- Reinhard Zumkeller, Sep 10 2013

a:=[1,3]; [1] cat [n le 2 select a[n] else 2*Self(n-1) + (2*n-2)*Self(n-2):n in [1..22]]; // Marius A. Burtea, Feb 12 2020

# Comment from the authors: For Maple program see A000903.
A000902 := n -> `if`(n=0, 1, I^(-n)*orthopoly[H](n, I)/2):
seq(A000902(n), n=0..22); # Peter Luschny, Nov 29 2017

n = 22; CoefficientList[ Series[(1/2)*(Exp[2*x+x^2] + 1), {x, 0, n}], x] * Table[k!, {k, 0, n}]
(* Jean-François Alcover, May 18 2011 *)
With[{nn=30},CoefficientList[Series[(Exp[2x+x^2]+1)/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 27 2025 *)

a(n) = 2*a(n-1) + (2n-2)*a(n-2) for n >= 3. - N. J. A. Sloane, Sep 23 2006
a(n) = 1 + n!/(2e) * [x^n] Sum[l>=0, 1/l! * {(1+x)^l-1}^2].
For asymptotics see the Robinson paper.
But the asymptotic formula in the Robinson paper is wrong (see A000898, discussion from Oct 01 2013). - Vaclav Kotesovec, Aug 04 2014
a(n) ~ 2^(n/2-3/2) * n^(n/2) * exp(sqrt(2*n)-n/2-1/2). - Vaclav Kotesovec, Aug 04 2014
a(n) = (i/2)^(1 - n)*KummerU((1 - n)/2, 3/2, -1) for n>=1. - Peter Luschny, Nov 29 2017
a(n) = Sum_{r=0..floor(n/2)} 2^(n-2r-1) * {(n!)/(r!(n-2r)!)}. - Aram Bingham, Feb 08 2020

A000901 Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).

Original entry on oeis.org

0, 0, 7, 74, 882, 11144, 159652, 2571960, 46406392, 928734944, 20436096048, 490489794464, 12752891909920, 357081983435904, 10712466529388608, 342798976818878336, 11655165558112403328, 419585962575107694080

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

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. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
R. G. Wilson, v, Comments on the Larsen paper (no date)

Maple
```
For Maple program see A000903.
```

For asymptotics see the Robinson paper.

Corrected and extended by Sean A. Irvine, Aug 23 2011

A122670 If n mod 4 = 2 or n mod 4 = 3 then a(n) = 0 else let m=floor(n/4), then a(n) = (2*m)!/m!.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 12, 12, 0, 0, 120, 120, 0, 0, 1680, 1680, 0, 0, 30240, 30240, 0, 0, 665280, 665280, 0, 0, 17297280, 17297280, 0, 0, 518918400, 518918400, 0, 0, 17643225600, 17643225600, 0, 0, 670442572800, 670442572800, 0, 0, 28158588057600, 28158588057600, 0, 0, 1295295050649600

Offset: 0

N. J. A. Sloane, Sep 23 2006

nonn

Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

A037224 is an essentially identical sequence.

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

If the duplicates and zeros are omitted we get A001813.

Cf. A000898, A000899, A000900, A000901, A000902, A000903, A037224.

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;
For Maple program see A000903.

Table[If[MemberQ[{2,3},Mod[n,4]],0,((2Floor[n/4])!/Floor[n/4]!)],{n,0,50}] (* Harvey P. Dale, Dec 30 2023 *)

For asymptotics see the Robinson paper.

a(n) = (1/2 + (-1)^(n/2 - 1/4 + (-1)^n/4)/2) * ((n/2 - 3/4 + (-1)^n/4 + (-1)^(n/2 - 1/4 + (-1)^n/4)/2)! / ((n/4 - 3/8 + (-1)^n/8 + (-1)^(n/2 - 1/4 + (-1)^n/4)/4)!)). - Wesley Ivan Hurt, Mar 30 2015

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

A122693 Bishops on an n X n board (see Robinson paper for details).

Original entry on oeis.org

0, 2, 4, 32, 128, 1152, 6912, 73728, 589824, 7372800, 73728000, 1061683200, 12740198400, 208089907200, 2913258700800, 53271016243200, 852336259891200, 17259809262796800, 310676566730342400, 6903923705118720000, 138078474102374400000, 3341499073277460480000

Offset: 0

N. J. A. Sloane, Sep 25 2006

nonn

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (C_{2n+1}, Eq. (20))

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

C:=proc(n) local k; if n mod 2 = 0 then RETURN(0); fi; k:=(n-1)/2; if k mod 2 = 0 then RETURN( k*2^(k-1)*((k/2)!)^2 ); else RETURN( 2^k*(((k+1)/2)!)^2 ); fi; end; [seq(C(2*n+1),n=0..30)];

c[n_] := Module[{k}, If[Mod[n, 2] == 0, 0, k = (n-1)/2; If[Mod[k, 2] == 0, k*2^(k-1)*((k/2)!)^2, 2^k*(((k+1)/2)!)^2]]];
a[n_] := c[2n+1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 23 2022, after Maple code *)

cp's OEIS Frontend

A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

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A000902 Expansion of e.g.f. (1/2)*(exp(2*x + x^2) + 1).

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A000901 Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).

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A122670 If n mod 4 = 2 or n mod 4 = 3 then a(n) = 0 else let m=floor(n/4), then a(n) = (2*m)!/m!.

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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

Mathematica

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

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Author

Keywords

References

Links

Programs

Maple

Mathematica

Extensions

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

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A000902 Expansion of e.g.f. (1/2)(exp(2x + x^2) + 1).