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

0,2

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A122370 A088349 A272802 * A327557 A163611 A160906

Adjacent sequences: A083063 A083064 A083065 * A083067 A083068 A083069

Keyword

nonn,easy

Author

Paul Barry, Apr 21 2003

Status

approved