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

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

Author

N. J. A. Sloane, Sep 28 2006

Keywords

• nonn

Links

• 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)

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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