A210343 -id:A210343 - 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

A210341 Triangle generated by T(n,k) = Fibonacci(n-k+2)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 9, 8, 1, 1, 8, 25, 27, 16, 1, 1, 13, 64, 125, 81, 32, 1, 1, 21, 169, 512, 625, 243, 64, 1, 1, 34, 441, 2197, 4096, 3125, 729, 128, 1, 1, 55, 1156, 9261, 28561, 32768, 15625, 2187, 256, 1, 1, 89, 3025, 39304, 194481, 371293

Offset: 0

Emanuele Munarini, Mar 20 2012

Number of tilings of an nXk chessboard using monomers and dimers of a fixed orientation. This is easy to see because the board here consists of k independent strips of length n. - Ralf Stephan, May 22 2014
Row sums = A210342
Central coefficients = A067966.
This triangle is related to the infinite Vandermonde matrix
V = [F(i+2)^j]_(i,j>=0) generated by Fibonacci numbers:
  1, 1,  1,   1,    1,      1,       1
  1, 2,  4,   8,    16,     32,      64
  1, 3,  9,   27,   81,     243,     729
  1, 5,  25,  125,  625,    3125,    15625
  1, 8,  64,  512,  4096,   32768,   262144
  1, 13, 169, 2197, 28561,  371293,  4826809
 1, 21, 441, 9261, 194481, 4084101, 85766121
The generating series of the columns can be expressed in terms of Fibonomial coefficients (A010048) (see Riordan's paper).

			Triangle begins:
  1
  1,  1
  1,  2,   1
  1,  3,   4,    1
  1,  5,   9,    8,    1
  1,  8,  25,   27,   16,    1
  1, 13,  64,  125,   81,   32,   1
  1, 21, 169,  512,  625,  243,  64,   1
  1, 34, 441, 2197, 4096, 3125, 729, 128, 1

Vincenzo Librandi, Rows n = 0..90, flattened
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

Cf. A103323, A067966, A210342, A210343.

[Fibonacci(n-k+2)^k: k in [0..n], n in [0..10]]; /* Alternatively: */ [[Fibonacci(n-k+2)^k: k in [0..n]]: n in [0..8]]; // Bruno Berselli, Mar 28 2012

Flatten[Table[Fibonacci[n-k+2]^k,{n,0,20},{k,0,n}]]

create_list(fib(n-k+2)^k,n,0,20,k,0,n);

G.f.: Sum_{k>=0} x^k/(1-Fibonacci(k+2)*x*y).

A182148 a(n) = Fibonacci(n-1)^n.

Original entry on oeis.org

1, 0, 1, 1, 16, 243, 15625, 2097152, 815730721, 794280046581, 2064377754059776, 13931233916552734375, 246990403565262140303521, 11447545997288281555215581184, 1389897885974444705448234373058929, 441692732032956477538220683055593208393

Offset: 0

Vincenzo Librandi, Jul 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..70

Cf. A000045, A100399, A210343.

Magma
```
[Fibonacci(n-1)^n: n in [0..15]];
```
Mathematica
```
Table[Fibonacci[n-1]^n,{n,0,20}]
```

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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A210341 Triangle generated by T(n,k) = Fibonacci(n-k+2)^k.

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A182148 a(n) = Fibonacci(n-1)^n.

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Magma

Mathematica