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 A069378 #21 Jan 07 2023 12:44:58
%S A069378 7,37,197,1041,5503,29089,153769,812849,4296863,22713981,120070149,
%T A069378 634712209,3355201895,17736195433,93756691401,495614587553,
%U A069378 2619907077991,13849295944501,73209847696773
%N A069378 Number of n X 3 binary arrays with a path of adjacent 1's from top row to bottom row.
%H A069378 G. C. Greubel, <a href="/A069378/b069378.txt">Table of n, a(n) for n = 1..1000</a>
%F A069378 G.f.: x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4+2*x^5). - _Vladeta Jovovic_, Jul 02 2003
%t A069378 Rest[CoefficientList[Series[x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4 +2*x^5), {x,0,50}], x]] (* _G. C. Greubel_, Apr 22 2018 *)
%o A069378 (PARI) x='x+O('x^30); Vec(x*(7-12*x+x^2+2*x^3-2*x^4)/(1 -7*x+9*x^2 +x^3- 4*x^4+2*x^5)) \\ _G. C. Greubel_, Apr 22 2018
%o A069378 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4+2*x^5))); // _G. C. Greubel_, Apr 22 2018
%Y A069378 Column 3 of A359576.
%Y A069378 Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, 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.
%K A069378 nonn
%O A069378 1,1
%A A069378 _R. H. Hardin_, Mar 22 2002