A067963 -id:A067963 - cp's OEIS frontend

A067966 Number of binary arrangements without adjacent 1's on n X n array connected n-s.

1, 2, 9, 125, 4096, 371293, 85766121, 52523350144, 83733937890625, 350356403707485209, 3833759992447475122176, 109879109551310452512114617, 8243206936713178643875538610721, 1619152874321527556575810000000000000

Offset: 0

R. H. Hardin, Feb 02 2002

Central coefficients of triangle A210341.

			Neighbors for n=4:
o o o o
| | | |
| | | |
o o o o
| | | |
| | | |
o o o o
| | | |
| | | |
o o o o

Vincenzo Librandi, Table of n, a(n) for n = 0..60
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69, 380.

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, 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, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

Cf. A100399, A210343, A210341.

[Fibonacci(n+2)^n: n in [0..13]]; // Bruno Berselli, Mar 28 2012

Mathematica
```
Table[Fibonacci[n+2]^n, {n, 0, 100}]
```
Maxima
```
makelist(fib(n+2)^n, n, 0, 14);
```

a(n)=fibonacci(n+2)^n \\ Charles R Greathouse IV, Mar 28 2012

a(n) = F(n+2)^n, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) ~ phi^2/sqrt(5) phi^n^2. [Charles R Greathouse IV, Mar 28 2012]

Edited by Dean Hickerson, Feb 15 2002

A067961 Number of binary arrangements without adjacent 1's on n X n torus connected n-s.

Original entry on oeis.org

1, 9, 64, 2401, 161051, 34012224, 17249876309, 23811286661761, 84590643846578176, 792594609605189126649, 19381341794579313317802199, 1242425797286480951825250390016, 208396491430277954192889648311785961, 91534759488004239323168528670973468727049

Offset: 1

R. H. Hardin, Feb 02 2002

			Neighbors for n=4:
| | | |
o o o o
| | | |
| | | |
o o o o
| | | |
| | | |
o o o o
| | | |
| | | |
o o o o
| | | |

Vincenzo Librandi, Table of n, a(n) for n = 1..69
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 409.

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.
Cf. A156216. - Paul D. Hanna, Sep 13 2010
Cf. A215941.

[Lucas(n)^n: n in [1..15]]; // Vincenzo Librandi, Mar 15 2014

a:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]^n:
seq(a(n), n=1..15);  # Alois P. Heinz, Aug 01 2021

Table[LucasL[n]^n,{n,15}] (* Harvey P. Dale, Mar 13 2014 *)

a(n) = L(n)^n, where L(n) = A000032(n) is the n-th Lucas number.
Logarithmic derivative of A156216. - Paul D. Hanna, Sep 13 2010
Sum_{n>=1} 1/a(n) = A215941. - Amiram Eldar, Nov 17 2020

Edited by Dean Hickerson, Feb 15 2002

A067965 Number of binary arrangements without adjacent 1's on n X n array connected ne-sw and nw-se.

Original entry on oeis.org

2, 9, 119, 2704, 177073, 21836929, 6985036032, 4576976735769, 7263963336910751, 24830487842030082304, 198126078679714777857441, 3494153303407491549112098721, 141264727800378056245286463971328, 12779122891585386852029424628087941481, 2628141044813862018744988536642011269669959

Offset: 1

R. H. Hardin, Feb 02 2002

			Neighbors for n=4 (dots represent spaces):
. o..o..o..o
...\/ \/ \/
.../\ /\ /\
. o..o..o..o
...\/ \/ \/
.../\ /\ /\
. o..o..o..o
...\/ \/ \/
.../\ /\ /\
. o..o..o..o

Liang Kai, Table of n, a(n) for n = 1..27 (first 19 terms from Vaclav Kotesovec)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69, 417.

Main diagonal of A181212.
Cf. circle A000204, line A000045, arrays: 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, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.
Cf. A201861, A212271.

Term a(14) from Vaclav Kotesovec, Dec 06 2011
Term a(15) from Vaclav Kotesovec, Jan 03 2012
Term a(16) from Vaclav Kotesovec, May 01 2012
Term a(17)-a(18) from Vaclav Kotesovec, Aug 13 2016

A067960 Number of binary arrangements without adjacent 1's on n X n torus connected ne-sw nw-se.

Original entry on oeis.org

1, 9, 34, 961, 25531, 2722500, 464483559, 224546142769, 215560806324388, 509113406167679889, 2590618817013278596997, 30737628149641669227004804, 809724336154415150287031740151, 48754690373355654118816600200711441

Offset: 1

R. H. Hardin, Feb 02 2002

If n is odd then A067960(n) = A027683(n).
a(18) = 2184710661251680812138610069332410066909052859790416601664. (a(17) = ?) - Vaclav Kotesovec, Sep 16 2014
a(20) = 61548416926224234005237372092957872593295040887178016957765412173582481. - Vaclav Kotesovec, May 18 2021

			Neighbors for n=4 (dots represent spaces):
.  \ /\ /\ /\ /
.   o..o..o..o
.  / \/ \/ \/ \
.  \ /\ /\ /\ /
.   o..o..o..o
.  / \/ \/ \/ \
.  \ /\ /\ /\ /
.   o..o..o..o
.  / \/ \/ \/ \
.  \ /\ /\ /\ /
.   o..o..o..o
.  / \/ \/ \/ \

Vaclav Kotesovec, Table of n, a(n) for n = 1..16
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 440.

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 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. A212271.

Terms a(12)-a(16) from Vaclav Kotesovec, May 18 2012

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.

Original entry on oeis.org

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

Offset: 0

R. H. Hardin, Feb 02 2002

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.

			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

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.

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.

a067962 n = a067962_list !! n
a067962_list = 1 : zipWith (*) a067962_list (drop 2 a001654_list)
-- Reinhard Zumkeller, Sep 24 2015

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

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

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

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

Edited by Dean Hickerson, Feb 15 2002

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

A228683 T(n,k)=Number of nXk binary arrays with no two ones adjacent horizontally, diagonally or antidiagonally.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 8, 19, 17, 16, 13, 40, 77, 41, 32, 21, 97, 216, 313, 99, 64, 34, 217, 809, 1152, 1277, 239, 128, 55, 508, 2529, 6737, 6160, 5215, 577, 256, 89, 1159, 8832, 28977, 56549, 32928, 21305, 1393, 512, 144, 2683, 28793, 152048, 333517, 475809, 176032

Offset: 1

R. H. Hardin Aug 30 2013

Table starts
...2....3......5.......8........13.........21...........34............55
...4....7.....19......40........97........217..........508..........1159
...8...17.....77.....216.......809.......2529.........8832.........28793
..16...41....313....1152......6737......28977.......152048........699833
..32...99...1277....6160.....56549.....333517......2644336......17124415
..64..239...5215...32928....475809....3837761.....46125216.....419022831
.128..577..21305..176032...4008817...44171841....806190208...10258304689
.256.1393..87049..941056..33795201..508425617..14105294112..251170142257
.512.3363.355685.5030848.284980061.5852202757.246929287360.6150224353031

			Some solutions for n=4 k=4
..0..0..0..1....1..0..0..0....0..0..1..0....0..0..1..0....1..0..0..0
..0..1..0..1....1..0..0..0....0..0..1..0....0..0..1..0....1..0..0..0
..0..0..0..0....0..0..0..1....1..0..1..0....0..0..1..0....1..0..0..0
..0..0..0..1....0..1..0..1....1..0..0..0....0..0..1..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1057

Column 1 is A000079
Column 2 is A001333(n+1)
Diagonal is A067963
Row 1 is A000045(n+2)
Row 2 is A006130(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1) +a(n-2)
k=3: a(n) = 5*a(n-1) -3*a(n-2) -3*a(n-3)
k=4: a(n) = 6*a(n-1) -2*a(n-2) -8*a(n-3)
k=5: a(n) = 12*a(n-1) -27*a(n-2) -32*a(n-3) +49*a(n-4) +20*a(n-5) -5*a(n-6)
k=6: [order 7]
k=7: [order 12]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2)
n=2: a(n) = a(n-1) +3*a(n-2)
n=3: a(n) = 2*a(n-1) +6*a(n-2) -5*a(n-3)
n=4: a(n) = 2*a(n-1) +16*a(n-2) -7*a(n-3) -18*a(n-4)
n=5: [order 7]
n=6: [order 10]
n=7: [order 16]

A067958 Number of binary arrangements without adjacent 1's on n X n torus connected e-w ne-sw n-s nw-se.

Original entry on oeis.org

1, 5, 10, 133, 1411, 42938, 1796859, 157763829, 22909432780, 6291183426165, 3032485231813445, 2674030233698391466, 4216437656471537450175, 12038380931111061789962901, 61810608197507432888286102310, 572863067272579464080483552434421

Offset: 1

R. H. Hardin, Feb 02 2002

For n > 1, a(n) is also the number of ways to populate an n X n toroidal chessboard with non-attacking kings (including the case of zero kings). - Vaclav Kotesovec, Oct 10 2011

			Neighbors for n=4:
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\
  :\|/\|/\|/\|/
  :-o--o--o--o-
  :/|\/|\/|\/|\

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 214.

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, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

Cf. A212269.

a(14) from Vaclav Kotesovec, Aug 22 2016

a(15)-a(16) from Vaclav Kotesovec, May 15 2021

A067964 Number of binary arrangements without adjacent 1's on n X n array connected n-s nw-se.

Original entry on oeis.org

2, 8, 90, 1876, 103484, 11462588, 3118943536, 1808994829500, 2465526600093372, 7394315828592829424, 50975951518289853305508, 784977037926751747674903856, 27509351187362150581313065415008, 2167705218542258344490649896364635660, 387057670485382113845659790427906287869964

Offset: 1

R. H. Hardin, Feb 02 2002

			Neighbors for n=4 (dots represent spaces):
. o..o..o..o
. |\ |\ |\ |
. | \| \| \|
. o..o..o..o
. |\ |\ |\ |
. | \| \| \|
. o..o..o..o
. |\ |\ |\ |
. | \| \| \|
. o..o..o..o

Vaclav Kotesovec, Table of n, a(n) for n = 1..21
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69-71.

Cf. circle A000204, line A000045, arrays: ne-sw nw-se A067965, e-w ne-sw nw-se A067963, 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, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

Limit n->infinity (a(n))^(1/n^2) = 1.503048082... (see A085850)

Terms a(14)-a(18) from Vaclav Kotesovec, May 01 2012

A067959 Number of binary arrangements without adjacent 1's on n X n torus connected ne-sw n-s nw-se.

Original entry on oeis.org

1, 7, 22, 547, 9021, 812830, 70046159, 24082448515, 10363980496342, 14228018243052057, 29400555005986658803, 166705587265151114516638, 1606507128309318588452521527, 38505096862341023166325442747581, 1696028983502674228038462924646464012

Offset: 1

R. H. Hardin, Feb 02 2002

			Neighbors for n=4 (dots represent spaces):
.\|/\|/\|/\|/
. o..o..o..o
./|\/|\/|\/|\
.\|/\|/\|/\|/
. o..o..o..o
./|\/|\/|\/|\
.\|/\|/\|/\|/
. o..o..o..o
./|\/|\/|\/|\
.\|/\|/\|/\|/
. o..o..o..o
./|\/|\/|\/|\

Sean A. Irvine, Table of n, a(n) for n = 1..17
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 73.

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, e-w ne-sw n-s nw-se A067958, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

a(13) from Vaclav Kotesovec, Aug 22 2016
a(14) from Vaclav Kotesovec, May 24 2021
a(15) from Sean A. Irvine, Jan 14 2024

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

Original entry on oeis.org

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000

Offset: 1

Vaclav Kotesovec, May 05 2012

nonn

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
V. Kotesovec, Non-attacking chess pieces

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Table[If[EvenQ[n],Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4,{j,1,n/2-1}],Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4,{j,1,(n-1)/2}]],{n,1,20}]

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

cp's OEIS Frontend

A067966 Number of binary arrangements without adjacent 1's on n X n array connected n-s.

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A067961 Number of binary arrangements without adjacent 1's on n X n torus connected n-s.

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A067965 Number of binary arrangements without adjacent 1's on n X n array connected ne-sw and nw-se.

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A067960 Number of binary arrangements without adjacent 1's on n X n torus connected ne-sw nw-se.

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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.

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A228683 T(n,k)=Number of nXk binary arrays with no two ones adjacent horizontally, diagonally or antidiagonally.

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A067958 Number of binary arrangements without adjacent 1's on n X n torus connected e-w ne-sw n-s nw-se.

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A067964 Number of binary arrangements without adjacent 1's on n X n array connected n-s nw-se.