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A083065 4th row of number array A083064. 10
1, 4, 19, 94, 469, 2344, 11719, 58594, 292969, 1464844, 7324219, 36621094, 183105469, 915527344, 4577636719, 22888183594, 114440917969, 572204589844, 2861022949219, 14305114746094, 71525573730469, 357627868652344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Inverse binomial transform of A090040=1,5,28,164,. With mentioned a(n) = 5*a(n-1)-1 also recurrence a(n) = 6*a(n-1)-5*a(n-2). Linked to A131577=0,1,2,4,8,16, via A154383, A154407 and A154410 = 10*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 respec. [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).

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

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

f[n_]:=5^n; lst={}; Do[a=f[n]; Do[a-=f[m], {m, n-1, 1, -1}]; AppendTo[lst, a/5], {n, 1, 30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

PROG

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

CROSSREFS

Cf. A007583, A083066, A007051.

Sequence in context: A131552 A122369 A005978 * A137636 A027618 A278678

Adjacent sequences:  A083062 A083063 A083064 * A083066 A083067 A083068

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Apr 21 2003

STATUS

approved

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Last modified July 22 07:23 EDT 2019. Contains 325216 sequences. (Running on oeis4.)