This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A163029 #42 Apr 15 2024 05:01:08 %S A163029 6,28,144,730,3692,18666,94384,477264,2413346,12203374,61707810, %T A163029 312032874,1577831334,7978491800,40344192708,204005208738, %U A163029 1031576601204,5216289773894,26376789637884,133377373911160,674438554337506 %N A163029 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 A163029 R. H. Hardin, <a href="/A163029/b163029.txt">Table of n, a(n) for n = 1..100</a> %H A163029 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 A163029 Chaim Goodman-Strauss, <a href="/A163029/a163029.nb">Mma notebook to accompany the above document</a> %F A163029 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 A163029 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 A163029 For an alternative proof see the Goodman-Strauss links. - _N. J. A. Sloane_, May 22 2020 %Y A163029 Cf. A001333 ((n-1) X 2 arrays), A059021 (no path required). %Y A163029 Cf. also A163030, A163031, A163032, A163033, A163034, A163035, A163036. %K A163029 nonn %O A163029 1,1 %A A163029 _R. H. Hardin_, Jul 20 2009