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.

1, 2, 12, 180, 7200, 748800, 204422400, 145957593600, 272940700032000, 1336044726656640000, 17122749216831498240000, 574502481723130428948480000, 50464872497041500009263431680000, 11605406728144633757130311383449600000

Offset

0,2

Comments

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.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..68

Sergey Kitaev and Toufik Mansour, The problem of the pawns, arXiv:math/0305253 [math.CO], 2003; Annals of Combinatorics 8 (2004) 81-91.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69, 421.

Formula

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 the 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

Example

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

Maple

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

Mathematica

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

Prog

(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

Crossrefs

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

Keyword

nonn,nice

Author

R. H. Hardin, Feb 02 2002

Extensions

Edited by Dean Hickerson, Feb 15 2002

Revised by N. J. A. Sloane following comments from Benoit Cloitre, Nov 12 2003

Status

approved