A069361 - OEIS

%I #26 Jun 21 2023 05:53:07

%S 1,17,197,1985,18621,167337,1461797,12519345,105683341,882516857,

%T 7308428597,60131384705,492202181661,4012347269577,32599584662597,

%U 264152863210065,2135714594033581,17236446198921497,138901692341235797,1117982939085627425,8989229069675479101

%N Number of 3 X n binary arrays with a path of adjacent 1's from top row to bottom row.

%H G. C. Greubel, <a href="/A069361/b069361.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-58,16).

%F G.f.: x*(1+2*x)/((1-8*x)*(2*x^2-7*x+1)). - _Vladeta Jovovic_, Jul 02 2003

%F From Maksym Voznyy (voznyy(AT)mail.ru), Jul 25 2008: (Start)

%F a(n) = 15*a(n-1) - 58*a(n-2) + 16*a(n-3), where a(1)=1, a(2)=17, a(3)=197;

%F a(n) = 8^n + 1/sqrt(41)*4^(n+1)*((7+sqrt(41))^(-(n+1)) - (7-sqrt(41))^(-(n+1))). (End)

%F a(n) = 8^n - A186446(n). - _R. J. Mathar_, Jan 27 2020

%e The 17 binary arrays for n=2:

%e 01 10 01 10 01 10 01 10 01 10 11 11 11 11 11 11 11

%e 01 10 01 10 11 11 11 11 11 11 01 10 01 01 11 11 11

%e 01 10 11 11 01 10 10 01 11 11 01 10 11 11 01 10 11 - _R. J. Mathar_, Jun 21 2023

%t CoefficientList[Series[(-2 z - 1)/(16 z^3 - 58 z^2 + 15 z - 1), {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 24 2011 *)

%o (PARI) x='x+O('x^30); Vec(x*(1+2*x)/((1-8*x)*(2*x^2-7*x+1))) \\ _G. C. Greubel_, Apr 22 2018

%Y Row 3 of A359576.

%Y 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 nonn,easy

%O 1,2

%A _R. H. Hardin_, Mar 22 2002