A083067 6th row of number array A083064.

1, 6, 41, 286, 2001, 14006, 98041, 686286, 4804001, 33628006, 235396041, 1647772286, 11534406001, 80740842006, 565185894041, 3956301258286, 27694108808001, 193858761656006, 1357011331592041, 9499079321144286

Offset: 0

Author

Paul Barry, Apr 21 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A052934 - Paul Barry, Apr 30 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (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). [From Milan Janjic, Feb 21 2010]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-7).

Crossrefs

Cf. A083066, A083068.

Programs

Magma

[(5*7^n+1)/6: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

Mathematica

f[n_]:=7^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/7],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

Formula

a(n) = (5*7^n+1)/6.

G.f.: (1-2*x)/((1-7*x)*(1-x)).

E.g.f.: (5*exp(7*x)+exp(x))/6.

a(n) = 7*a(n-1)-1 with a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = 8*a(n-1)-7*a(n-2). - Vincenzo Librandi, Nov 06 2011

a(n) = 7^n - sum(7^i, i=0..n-1) for n>0. [Bruno Berselli, Jun 20 2013]