|
| |
|
|
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.
|
|
94
|
|
|
|
3, 16, 84, 440, 2304, 12064, 63168, 330752, 1731840, 9068032, 47480832, 248612864, 1301753856, 6816071680, 35689414656, 186872201216, 978475548672, 5123364487168, 26826284728320, 140464250421248, 735480363614208, 3851025180000256, 20164229625544704, 105581277033267200
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
| |
|
|
