A083065 4th row of number array A083064.

1, 4, 19, 94, 469, 2344, 11719, 58594, 292969, 1464844, 7324219, 36621094, 183105469, 915527344, 4577636719, 22888183594, 114440917969, 572204589844, 2861022949219, 14305114746094, 71525573730469, 357627868652344

Offset: 0

Author

Paul Barry, Apr 21 2003

Keywords

• nonn
• easy

Comments

Inverse binomial transform of A090040. [Paul Curtz, Jan 11 2009]

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=7, (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]

For an integer x, consider the sequence P(x) of polynomials p_{1}, p_{2}, p_{3}, . . . defined by p_{1} = x-1, p_{n+1} = x*p_{1} - 1. P(5) = This sequence. P(1), P(2), P(3), P(4) are A000004, A123412, A007051, A007583 resp. [K.V.Iyer, Jun 22 2010]

It appears that if s(n) is a first order rational sequence of the form s(0)=2, s(n)= (3*s(n-1)+2)/(2*s(n-1)+3), n>0, then s(n)=2*a(n)/(2*a(n)-1), n>0.

An Engel expansion of 5/3 to the base b := 5/4 as defined in A181565, with the associated series expansion 5/3 = b + b^2/4 + b^3/(4*19) + b^4/(4*19*94) + .... Cf. A007051. - Peter Bala, Oct 29 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-5).

Crossrefs

Cf. A007583, A083066, A007051.

Programs

Magma

[(3*5^n+1)/4: n in [0..30]]; // Vincenzo Librandi, Nov 04 2011

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*5-1 od: seq(a[n], n=1..22); # Zerinvary Lajos, Feb 22 2008

Mathematica

CoefficientList[Series[(1-2x)/((1-5x)(1-x)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-5},{1,4},30] (* Harvey P. Dale, Jul 27 2022 *)

Formula

a(n) = (3*5^n+1)/4.

G.f.: (1-2*x)/((1-5*x)(1-x)).

E.g.f.: (3*exp(5*x) + exp(x))/4.

a(n) = 5*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = 6*a(n-1)-5*a(n-2). - Vincenzo Librandi, Nov 04 2011

a(n) = 5^n - Sum_{i=0..n-1} 5^i. - Bruno Berselli, Jun 20 2013