A083066 5th row of number array A083064.

1, 5, 29, 173, 1037, 6221, 37325, 223949, 1343693, 8062157, 48372941, 290237645, 1741425869, 10448555213, 62691331277, 376147987661, 2256887925965, 13541327555789, 81247965334733, 487487792008397, 2924926752050381

Offset: 0

Author

Paul Barry, Apr 21 2003

Keywords

• nonn
• easy

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
• R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, eq (43).
• Index entries for linear recurrences with constant coefficients, signature (7,-6).

Crossrefs

Cf. A083064, A083065, A083067, A007051, A007583.

Programs

Magma

[(4*6^n+1)/5: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

Mathematica

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 *)

Formula

a(n) = (4*6^n+1)/5.

G.f.: (1-2*x)/((1-6*x)*(1-x)).

E.g.f.: (4*exp(6*x)+exp(x))/5.

a(n) = 6*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = 7*a(n-1)-6*a(n-2). - Vincenzo Librandi, Nov 04 2011

a(n) = 6^n - Sum_{i=0..n-1} 6^i for n>0. - Bruno Berselli, Jun 20 2013