A069429 Half the number of 3 X n binary arrays with no path of adjacent 1's or adjacent 0's from top row to bottom row.

3, 16, 84, 440, 2304, 12064, 63168, 330752, 1731840, 9068032, 47480832, 248612864, 1301753856, 6816071680, 35689414656, 186872201216, 978475548672, 5123364487168, 26826284728320, 140464250421248, 735480363614208, 3851025180000256, 20164229625544704, 105581277033267200

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (6,-4).

Formula

Empirical G.f.: x*(3-2*x)/(1-6*x+4*x^2). - Colin Barker, Feb 22 2012

Empirical: a(n) = 3*A084326(n) - 2*A084326(n-1). - R. J. Mathar, Nov 09 2018

From Andrew Howroyd, Oct 27 2020: (Start)

The above conjectures are true and follow from formulas given in A069361 and A069396.

a(n) = (8^n)/2 - A069361(n) + A069396(n).

a(n) = 2^(n-1)*Fibonacci(2*n+2) = A084326(n+1)/2. (End)

Example

From Andrew Howroyd, Oct 27 2020: (Start)
Some of the 2*a(2) = 32 arrays are:
  0 0   0 0   0 0   0 1   0 1   0 0   0 1
  0 0   0 1   1 1   1 0   1 0   1 1   1 0
  1 1   1 1   1 1   1 1   0 1   0 0   1 1
(End)

Mathematica

LinearRecurrence[{6, -4}, {3, 16}, 100] (* Jean-François Alcover, Nov 01 2020 *)

Prog

(PARI) Vec((3 - 2*x)/(1 - 6*x + 4*x^2) + O(x^30)) \\ Andrew Howroyd, Oct 27 2020
(PARI) a(n) = 2^(n-1)*fibonacci(2*n+2) \\ Andrew Howroyd, Oct 27 2020

Crossrefs

Cf. 2 X n A000079, n X 1 A000225, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

Cf. A084326.

Sequence in context: A041707 A037584 A030983 * A275402 A026131 A026160

Adjacent sequences: A069426 A069427 A069428 * A069430 A069431 A069432

Keyword

nonn,easy

Author

R. H. Hardin, Mar 22 2002

Extensions

Terms a(21) and beyond from Andrew Howroyd, Oct 27 2020

Status

approved