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%I #43 Mar 21 2023 15:59:49
%S 1,5,29,173,1037,6221,37325,223949,1343693,8062157,48372941,290237645,
%T 1741425869,10448555213,62691331277,376147987661,2256887925965,
%U 13541327555789,81247965334733,487487792008397,2924926752050381
%N 5th row of number array A083064.
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=8, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - _Milan Janjic_, Feb 21 2010
%C An Engel expansion of 3/2 to the base b := 6/5 as defined in A181565, with the associated series expansion 3/2 = b + b^2/5 + b^3/(5*29) + b^4/(5*29*173) + .... Cf. A007051. - _Peter Bala_, Oct 29 2013
%H Vincenzo Librandi, <a href="/A083066/b083066.txt">Table of n, a(n) for n = 0..1000</a>
%H Mudit Aggarwal and Samrith Ram, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Ram/ram3.html">Generating Functions for Straight Polyomino Tilings of Narrow Rectangles</a>, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1406.7788">Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices</a>, arXiv:1406.7788 [math.CO], 2014, eq (43).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-6).
%F a(n) = (4*6^n+1)/5.
%F G.f.: (1-2*x)/((1-6*x)*(1-x)).
%F E.g.f.: (4*exp(6*x)+exp(x))/5.
%F a(n) = 6*a(n-1)-1 with n>0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = 7*a(n-1)-6*a(n-2). - _Vincenzo Librandi_, Nov 04 2011
%F a(n) = 6^n - Sum_{i=0..n-1} 6^i for n>0. - _Bruno Berselli_, Jun 20 2013
%t f[n_]:=6^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/6],{n,1,30}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 10 2010 *)
%o (Magma) [(4*6^n+1)/5: n in [0..30]]; // _Vincenzo Librandi_, Nov 06 2011
%Y Cf. A083064, A083065, A083067, A007051, A007583.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Apr 21 2003
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