A123071 - OEIS

%I #19 Jul 25 2022 08:52:23

%S 1,2,4,12,36,120,400,1520,5776,23712,97344,431808,1915456,9012608,

%T 42406144,210988800,1049760000,5475340800,28558296064,155672726528,

%U 848579961856,4810614454272,27271456395264,160376430784512,943132599095296,5735299537018880

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

%H R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence S(2k+1) eq(24) p. 210.]

%H R. W. Robinson, <a href="/A000899/a000899_1.pdf">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

%F 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

%p For Maple program see A005635.

%p # alternative

%p # this is A000898, replicated as 1,1,2,2,6,6,20,20,76,76,...

%p B := proc(n)

%p     if n=0 or n= -2 then

%p         1 ;

%p     elif type (n,'odd') then

%p         procname(n-1) ;

%p     else

%p         2*procname(n-2)+(n-2)*procname(n-4) ;

%p     end if;

%p end proc:

%p A123071 := proc(n)

%p     B(n)*B(n+1) ;

%p end proc:

%p seq(A123071(n),n=0..20) ; # _R. J. Mathar_, Apr 02 2017

%t 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]];

%t a[n_] := B[n]*B[n+1];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jul 23 2022, after _R. J. Mathar_ *)

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

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Sep 28 2006