%I #40 Nov 01 2020 04:10:33

%S 3,16,84,440,2304,12064,63168,330752,1731840,9068032,47480832,

%T 248612864,1301753856,6816071680,35689414656,186872201216,

%U 978475548672,5123364487168,26826284728320,140464250421248,735480363614208,3851025180000256,20164229625544704,105581277033267200

%N 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.

%H Andrew Howroyd, <a href="/A069429/b069429.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).

%F Empirical G.f.: x*(3-2*x)/(1-6*x+4*x^2). - _Colin Barker_, Feb 22 2012

%F Empirical: a(n) = 3*A084326(n) - 2*A084326(n-1). - _R. J. Mathar_, Nov 09 2018

%F From _Andrew Howroyd_, Oct 27 2020: (Start)

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

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

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

%e From _Andrew Howroyd_, Oct 27 2020: (Start)

%e Some of the 2*a(2) = 32 arrays are:

%e 0 0 0 0 0 0 0 1 0 1 0 0 0 1

%e 0 0 0 1 1 1 1 0 1 0 1 1 1 0

%e 1 1 1 1 1 1 1 1 0 1 0 0 1 1

%e (End)

%t LinearRecurrence[{6, -4}, {3, 16}, 100] (* _Jean-François Alcover_, Nov 01 2020 *)

%o (PARI) Vec((3 - 2*x)/(1 - 6*x + 4*x^2) + O(x^30)) \\ _Andrew Howroyd_, Oct 27 2020

%o (PARI) a(n) = 2^(n-1)*fibonacci(2*n+2) \\ _Andrew Howroyd_, Oct 27 2020

%Y 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.

%Y Cf. A084326.

%K nonn,easy

%O 1,1

%A _R. H. Hardin_, Mar 22 2002

%E Terms a(21) and beyond from _Andrew Howroyd_, Oct 27 2020