%I #42 Apr 15 2024 05:01:08
%S 6,28,144,730,3692,18666,94384,477264,2413346,12203374,61707810,
%T 312032874,1577831334,7978491800,40344192708,204005208738,
%U 1031576601204,5216289773894,26376789637884,133377373911160,674438554337506
%N Number of n X 3 binary arrays with all 1's connected and a path of 1's from top row to bottom row.
%H R. H. Hardin, <a href="/A163029/b163029.txt">Table of n, a(n) for n = 1..100</a>
%H Chaim Goodman-Strauss, <a href="/A163029/a163029.pdf">Notes on the number of m × n binary arrays with all 1’s connected and a path of 1’s from top row to bottom row</a> (May 21 2020)
%H Chaim Goodman-Strauss, <a href="/A163029/a163029.nb">Mma notebook to accompany the above document</a>
%F a(n) = 7*a(n-1) - 11*a(n-2) + 6*a(n-3) + a(n-4) - 7*a(n-5) + a(n-6). [Conjectured by _R. J. Mathar_, Aug 11 2009]
%F Proof from _Peter Kagey_, May 08 2019: Scanning from top to bottom, there are 6 possible intermediate states that the bottom row can be in. The transitions between these states define a 6 X 6 transition matrix whose characteristic polynomial agrees with the characteristic polynomial of the above recurrence. QED
%F For an alternative proof see the Goodman-Strauss links. - _N. J. A. Sloane_, May 22 2020
%Y Cf. A001333 ((n-1) X 2 arrays), A059021 (no path required).
%Y Cf. also A163030, A163031, A163032, A163033, A163034, A163035, A163036.
%K nonn
%O 1,1
%A _R. H. Hardin_, Jul 20 2009