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A067962 a(n) = F(n+2)*prod(i=1,n+1,F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number. 12


%S 1,2,12,180,7200,748800,204422400,145957593600,272940700032000,

%T 1336044726656640000,17122749216831498240000,

%U 574502481723130428948480000,50464872497041500009263431680000,11605406728144633757130311383449600000

%N a(n) = F(n+2)*prod(i=1,n+1,F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.

%C Number of binary arrangements without adjacent 1's on n X n array connected nw-se.

%C Kitaev and Mansour give a general formula for the number of binary m X n matrices avoiding certain configurations.

%H Reinhard Zumkeller, <a href="/A067962/b067962.txt">Table of n, a(n) for n = 0..68</a>

%H Sergey Kitaev and Toufik Mansour, <a href="http://arXiv.org/abs/math.CO/0305253">The problem of the pawns</a>, Annals of Combinatorics 8 (2004) 81-91.

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 69, 421.

%F 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 a(n) is asymptotic to C^2*((1+sqrt(5))/2)^((n+2)^2)/(5^(n+3/2)) where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073. - _Vaclav Kotesovec_, Oct 28 2011

%F a(n) = a(n-1) * A001654(n+1), n > 0. - _Reinhard Zumkeller_, Sep 24 2015

%e Neighbors for n=4 (dots represent spaces, circles represent grid points):

%e O..O..O..O

%e .\..\..\..

%e ..\..\..\.

%e O..O..O..O

%e .\..\..\..

%e ..\..\..\.

%e O..O..O..O

%e .\..\..\..

%e ..\..\..\.

%e O..O..O..O

%p a:= proc(n) option remember; `if`(n=0, 1, (F->

%p F(n+1)*F(n+2)*a(n-1))(combinat[fibonacci]))

%p end:

%p seq(a(n), n=0..14); # _Alois P. Heinz_, May 20 2019

%t Rest[Table[With[{c=Fibonacci[Range[n]]},(Times@@Most[c])^2 Last[c]],{n,15}]] (* _Harvey P. Dale_, Dec 17 2013 *)

%o (PARI) a(n)=fibonacci(n+2)*prod(i=0,n,fibonacci(i+1))^2

%o (Haskell)

%o a067962 n = a067962_list !! n

%o a067962_list = 1 : zipWith (*) a067962_list (drop 2 a001654_list)

%o -- _Reinhard Zumkeller_, Sep 24 2015

%Y 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 Cf. A001654, A003266.

%K nonn,nice

%O 0,2

%A _R. H. Hardin_, Feb 02 2002

%E Edited by _Dean Hickerson_, Feb 15 2002

%E Revised by _N. J. A. Sloane_ following comments from _Benoit Cloitre_, Nov 12 2003

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Last modified February 27 15:49 EST 2020. Contains 332307 sequences. (Running on oeis4.)