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%I #56 Oct 25 2023 09:41:02
%S 1,9,89,889,8889,88889,888889,8888889,88888889,888888889,8888888889,
%T 88888888889,888888888889,8888888888889,88888888888889,
%U 888888888888889,8888888888888889,88888888888888889,888888888888888889,8888888888888888889,88888888888888888889,888888888888888888889
%N a(0)=1, a(n) = a(n-1) + 8*10^(n-1).
%C 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
%C 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
%C Except for the initial term, these are the 9-automorphic numbers ending in 9. - _Eric M. Schmidt_, Aug 17 2012
%H Harry J. Smith, <a href="/A059482/b059482.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%F a(n) = (10^n)*(1000/1125) + (1/9).
%F a(n) = A002282(n) + 1 = (8*10^n + 1)/9.
%F a(n) = 10*a(n-1) - 1 with n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 07 2010
%F G.f.: -(2*x-1) / ((x-1)*(10*x-1)). - _Colin Barker_, Feb 02 2013
%F a(n) = 10^n - Sum_{i=0..n-1} 10^i for n > 0. - _Bruno Berselli_, Jun 20 2013
%F E.g.f.: exp(x)*(1 + 8*exp(9*x))/9. - _Stefano Spezia_, Oct 25 2023
%e a(3) = (10^3)*(1000/1125) + (1/9) = (8000/9) + (1/9) = 889.
%t Table[(8*10^n+1)/9, {n,0,50}] (* _G. C. Greubel_, May 15 2017 *)
%o (PARI) { a=1/5; for (n = 0, 200, a+=8*10^(n - 1); write("b059482.txt", n, " ", a); ) } \\ _Harry J. Smith_, Jun 27 2009
%o (Python) def a(n): return (8*10**n+1)//9 # _Martin Gergov_, Oct 20 2022
%Y Cf. A002282.
%K nonn,easy
%O 0,2
%A _Anton Joha_, Feb 04 2001
%E More terms from _Henry Bottomley_, Feb 05 2001
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