A059482 a(0)=1, a(n) = a(n-1) + 8*10^(n-1).

1, 9, 89, 889, 8889, 88889, 888889, 8888889, 88888889, 888888889, 8888888889, 88888888889, 888888888889, 8888888888889, 88888888888889, 888888888888889, 8888888888888889, 88888888888888889, 888888888888888889, 8888888888888888889, 88888888888888888889, 888888888888888888889

Offset: 0

Author

Anton Joha, Feb 04 2001

Keywords

• nonn
• easy

Comments

Related to the sum of Fibonacci-variants: Sum of the (Fibonacci numbers)/(10^n) = 0/(10^1) + 1/(10^2) + 1/(10^3) + 2/(10^4) + 3/(10^5) + 5/(10^6) + ... = 1/89. Sum of the (tribonacci numbers)/(10^(n+1)) = 1/889. Sum of the (tetranacci numbers)/(10^(n+2)) = 1/8889, etc. The denominators of those sums is the current sequence. The first one is of course 0.11111111111... = 1/9. - partially edited by Michel Marcus, Jan 27 2014

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=12, (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

Except for the initial term, these are the 9-automorphic numbers ending in 9. - Eric M. Schmidt, Aug 17 2012

Examples

			a(3) = (10^3)*(1000/1125) + (1/9) = (8000/9) + (1/9) = 889.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (11,-10).

Crossrefs

Cf. A002282.

Programs

Mathematica

Table[(8*10^n+1)/9, {n,0,50}] (* G. C. Greubel, May 15 2017 *)

PARI

{ a=1/5; for (n = 0, 200, a+=8*10^(n - 1); write("b059482.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 27 2009

Python

def a(n): return (8*10**n+1)//9 # Martin Gergov, Oct 20 2022

Formula

a(n) = (10^n)*(1000/1125) + (1/9).

a(n) = A002282(n) + 1 = (8*10^n + 1)/9.

a(n) = 10*a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 07 2010

G.f.: -(2*x-1) / ((x-1)*(10*x-1)). - Colin Barker, Feb 02 2013

a(n) = 10^n - Sum_{i=0..n-1} 10^i for n > 0. - Bruno Berselli, Jun 20 2013

E.g.f.: exp(x)*(1 + 8*exp(9*x))/9. - Stefano Spezia, Oct 25 2023

Extensions

More terms from Henry Bottomley, Feb 05 2001