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A067962 a(n) = F(n+2)*(Product_{i=1..n+1} F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.

%I A067962 #41 Jan 09 2021 11:05:48
%S A067962 1,2,12,180,7200,748800,204422400,145957593600,272940700032000,
%T A067962 1336044726656640000,17122749216831498240000,
%U A067962 574502481723130428948480000,50464872497041500009263431680000,11605406728144633757130311383449600000
%N A067962 a(n) = F(n+2)*(Product_{i=1..n+1} F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.
%C A067962 Number of binary arrangements without adjacent 1's on n X n array connected nw-se.
%C A067962 Kitaev and Mansour give a general formula for the number of binary m X n matrices avoiding certain configurations.
%H A067962 Reinhard Zumkeller, <a href="/A067962/b067962.txt">Table of n, a(n) for n = 0..68</a>
%H A067962 Sergey Kitaev and Toufik Mansour, <a href="https://arxiv.org/abs/math/0305253">The problem of the pawns</a>, arXiv:math/0305253 [math.CO], 2003; Annals of Combinatorics 8 (2004) 81-91.
%H A067962 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 69, 421.
%F A067962 a(n) = (F(3) * F(4) * ... * F(n+1))^2 * F(n+2), where F(n) = A000045(n) is the n-th Fibonacci number.
%F A067962 a(n) is asymptotic to C^2*((1+sqrt(5))/2)^((n+2)^2)/(5^(n+3/2)) where C=1.226742010720353244... is the Fibonacci Factorial Constant, see A062073. - _Vaclav Kotesovec_, Oct 28 2011
%F A067962 a(n) = a(n-1) * A001654(n+1), n > 0. - _Reinhard Zumkeller_, Sep 24 2015
%e A067962 Neighbors for n=4 (dots represent spaces, circles represent grid points):
%e A067962 O..O..O..O
%e A067962 .\..\..\..
%e A067962 ..\..\..\.
%e A067962 O..O..O..O
%e A067962 .\..\..\..
%e A067962 ..\..\..\.
%e A067962 O..O..O..O
%e A067962 .\..\..\..
%e A067962 ..\..\..\.
%e A067962 O..O..O..O
%p A067962 a:= proc(n) option remember; `if`(n=0, 1, (F->
%p A067962       F(n+1)*F(n+2)*a(n-1))(combinat[fibonacci]))
%p A067962     end:
%p A067962 seq(a(n), n=0..14);  # _Alois P. Heinz_, May 20 2019
%t A067962 Rest[Table[With[{c=Fibonacci[Range[n]]},(Times@@Most[c])^2 Last[c]],{n,15}]] (* _Harvey P. Dale_, Dec 17 2013 *)
%o A067962 (PARI) a(n)=fibonacci(n+2)*prod(i=0,n,fibonacci(i+1))^2
%o A067962 (Haskell)
%o A067962 a067962 n = a067962_list !! n
%o A067962 a067962_list = 1 : zipWith (*) a067962_list (drop 2 a001654_list)
%o A067962 -- _Reinhard Zumkeller_, Sep 24 2015
%Y A067962 Cf. circle A000204, line A000045, arrays: ne-sw nw-se A067965, e-w ne-sw nw-se A067963, n-s nw-se A067964, e-w n-s nw-se A066864, e-w ne-sw n-s nw-se A063443, n-s A067966, e-w n-s A006506, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, e-w ne-sw n-s nw-se A067958, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.
%Y A067962 Cf. A001654, A003266.
%K A067962 nonn,nice
%O A067962 0,2
%A A067962 _R. H. Hardin_, Feb 02 2002
%E A067962 Edited by _Dean Hickerson_, Feb 15 2002
%E A067962 Revised by _N. J. A. Sloane_ following comments from _Benoit Cloitre_, Nov 12 2003