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%I #36 Feb 16 2022 23:08:04
%S 1,9,64,2401,161051,34012224,17249876309,23811286661761,
%T 84590643846578176,792594609605189126649,19381341794579313317802199,
%U 1242425797286480951825250390016,208396491430277954192889648311785961,91534759488004239323168528670973468727049
%N Number of binary arrangements without adjacent 1's on n X n torus connected n-s.
%H Vincenzo Librandi, <a href="/A067961/b067961.txt">Table of n, a(n) for n = 1..69</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 409.
%F a(n) = L(n)^n, where L(n) = A000032(n) is the n-th Lucas number.
%F Logarithmic derivative of A156216. - _Paul D. Hanna_, Sep 13 2010
%F Sum_{n>=1} 1/a(n) = A215941. - _Amiram Eldar_, Nov 17 2020
%e Neighbors for n=4:
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%e o o o o
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%e o o o o
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%e | | | |
%e o o o o
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%e o o o o
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%p a:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]^n:
%p seq(a(n), n=1..15); # _Alois P. Heinz_, Aug 01 2021
%t Table[LucasL[n]^n,{n,15}] (* _Harvey P. Dale_, Mar 13 2014 *)
%o (Magma) [Lucas(n)^n: n in [1..15]]; // _Vincenzo Librandi_, Mar 15 2014
%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, nw-se A067962, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, e-w ne-sw n-s nw-se A067958, e-w n-s A027683, e-w ne-sw n-s A066866.
%Y Cf. A156216. - _Paul D. Hanna_, Sep 13 2010
%Y Cf. A215941.
%K nonn,nice
%O 1,2
%A _R. H. Hardin_, Feb 02 2002
%E Edited by _Dean Hickerson_, Feb 15 2002
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