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A014832 a(1)=1; for n>1, a(n) = 9*a(n-1) + n.

%I A014832 #31 Mar 29 2025 18:00:24
%S A014832 1,11,102,922,8303,74733,672604,6053444,54481005,490329055,4412961506,
%T A014832 39716653566,357449882107,3217048938977,28953440450808,
%U A014832 260580964057288,2345228676515609,21107058088640499,189963522797764510,1709671705179880610,15387045346618925511,138483408119570329621
%N A014832 a(1)=1; for n>1, a(n) = 9*a(n-1) + n.
%H A014832 Seiichi Manyama, <a href="/A014832/b014832.txt">Table of n, a(n) for n = 1..1000</a>
%H A014832 Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, <a href="https://arxiv.org/abs/2501.06675">Chip-Firing on Infinite k-ary Trees</a>, arXiv:2501.06675 [math.CO], 2025. See p. 18.
%H A014832 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-19,9).
%F A014832 a(n) = (9^(n+1) - 8*n - 9)/64. - _Rolf Pleisch_, Oct 22 2010
%F A014832 From _Harvey P. Dale_, May 01 2012: (Start)
%F A014832 a(1)=1, a(2)=11, a(3)=102; for n>3, a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
%F A014832 G.f.: -x/((x-1)^2*(9*x-1)). (End)
%F A014832 a(n) = Sum_{i=0..n-1} 8^i*binomial(n+1,n-1-i). - _Bruno Berselli_, Nov 13 2015
%F A014832 E.g.f.: exp(x)*(9*exp(8*x) - 8*x - 9)/64. - _Elmo R. Oliveira_, Mar 29 2025
%e A014832 For n=5, a(5) = 1*15 + 8*20 + 8^2*15 + 8^3*6 + 8^4*1 = 8303. [_Bruno Berselli_, Nov 13 2015]
%p A014832 a:=n->sum((9^(n-j)-1)/8,j=0..n): seq(a(n), n=1..18); # _Zerinvary Lajos_, Jan 15 2007
%p A014832 a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,9]])^n)[2,3]: seq(a(n), n=1..18); # _Alois P. Heinz_, Aug 06 2008
%t A014832 RecurrenceTable[{a[1]==1,a[n]==9a[n-1]+n},a,{n,20}] (* or *) LinearRecurrence[ {11,-19,9},{1,11,102},20] (* _Harvey P. Dale_, May 01 2012 *)
%Y A014832 Cf. A001018, A104712.
%K A014832 nonn,easy
%O A014832 1,2
%A A014832 _N. J. A. Sloane_