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A067962
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a(n) = F(n+2)*prod(i=1,n+1,F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.
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12
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1, 2, 12, 180, 7200, 748800, 204422400, 145957593600, 272940700032000, 1336044726656640000, 17122749216831498240000, 574502481723130428948480000, 50464872497041500009263431680000, 11605406728144633757130311383449600000
(list;
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history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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Number of binary arrangements without adjacent 1's on n X n array connected nw-se.
Kitaev and Mansour give a general formula for the number of binary m X n matrices avoiding certain configurations.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..68
Sergey Kitaev and Toufik Mansour, The problem of the pawns, Annals of Combinatorics 8 (2004) 81-91.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69, 421.
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FORMULA
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a(n) = (F(3) * F(4) * ... * F(n+1))^2 * F(n+2), where F(n) = A000045(n) is the n-th Fibonacci number.
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
a(n) = a(n-1) * A001654(n+1), n > 0. - Reinhard Zumkeller, Sep 24 2015
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EXAMPLE
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Neighbors for n=4 (dots represent spaces, circles represent grid points):
O..O..O..O
.\..\..\..
..\..\..\.
O..O..O..O
.\..\..\..
..\..\..\.
O..O..O..O
.\..\..\..
..\..\..\.
O..O..O..O
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, (F->
F(n+1)*F(n+2)*a(n-1))(combinat[fibonacci]))
end:
seq(a(n), n=0..14); # Alois P. Heinz, May 20 2019
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MATHEMATICA
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Rest[Table[With[{c=Fibonacci[Range[n]]}, (Times@@Most[c])^2 Last[c]], {n, 15}]] (* Harvey P. Dale, Dec 17 2013 *)
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PROG
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(PARI) a(n)=fibonacci(n+2)*prod(i=0, n, fibonacci(i+1))^2
(Haskell)
a067962 n = a067962_list !! n
a067962_list = 1 : zipWith (*) a067962_list (drop 2 a001654_list)
-- Reinhard Zumkeller, Sep 24 2015
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CROSSREFS
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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.
Cf. A001654, A003266.
Sequence in context: A228508 A156516 A228593 * A134716 A243807 A006023
Adjacent sequences: A067959 A067960 A067961 * A067963 A067964 A067965
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KEYWORD
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nonn,nice
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AUTHOR
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R. H. Hardin, Feb 02 2002
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EXTENSIONS
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Edited by Dean Hickerson, Feb 15 2002
Revised by N. J. A. Sloane following comments from Benoit Cloitre, Nov 12 2003
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STATUS
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approved
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