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
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Apr 21 2003
STATUS
approved