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

0,2

Links

Table of n, a(n) for n=0..25.

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)

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

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 *)

Crossrefs

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

Sequence in context: A148208 A245798 A241530 * A048116 A148209 A148210

Adjacent sequences: A123068 A123069 A123070 * A123072 A123073 A123074

Keyword

nonn

Author

N. J. A. Sloane, Sep 28 2006

Status

approved