A014832 a(1)=1; for n>1, a(n) = 9*a(n-1) + n.
1, 11, 102, 922, 8303, 74733, 672604, 6053444, 54481005, 490329055, 4412961506, 39716653566, 357449882107, 3217048938977, 28953440450808, 260580964057288, 2345228676515609, 21107058088640499, 189963522797764510, 1709671705179880610, 15387045346618925511, 138483408119570329621
Offset: 1
Examples
For n=5, a(5) = 1*15 + 8*20 + 8^2*15 + 8^3*6 + 8^4*1 = 8303. [_Bruno Berselli_, Nov 13 2015]
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..1000
- Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 18.
- Index entries for linear recurrences with constant coefficients, signature (11,-19,9).
Programs
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Maple
a:=n->sum((9^(n-j)-1)/8,j=0..n): seq(a(n), n=1..18); # Zerinvary Lajos, Jan 15 2007 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
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Mathematica
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 *)
Formula
a(n) = (9^(n+1) - 8*n - 9)/64. - Rolf Pleisch, Oct 22 2010
From Harvey P. Dale, May 01 2012: (Start)
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).
G.f.: -x/((x-1)^2*(9*x-1)). (End)
a(n) = Sum_{i=0..n-1} 8^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
E.g.f.: exp(x)*(9*exp(8*x) - 8*x - 9)/64. - Elmo R. Oliveira, Mar 29 2025