A014830 a(1)=1; for n > 1, a(n) = 7*a(n-1) + n.
1, 9, 66, 466, 3267, 22875, 160132, 1120932, 7846533, 54925741, 384480198, 2691361398, 18839529799, 131876708607, 923136960264, 6461958721864, 45233711053065, 316635977371473, 2216451841600330, 15515162891202330, 108606140238416331, 760242981668914339, 5321700871682400396
Offset: 1
Examples
For n=5, a(5) = 1*15 + 6*20 + 6^2*15 + 6^3*6 + 6^4*1 = 3267. - _Bruno Berselli_, Nov 13 2015
Links
- Colin Barker, 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 (9,-15,7).
Programs
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Maple
a:=n->sum((7^(n-j)-1)/6,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007
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Mathematica
a[1] = 1; a[n_] := 7*a[n-1]+n; Table[a[n], {n, 10}] (* Zak Seidov, Feb 06 2011 *) LinearRecurrence[{9, -15, 7}, {1, 9, 66}, 30] (* Harvey P. Dale, Jul 22 2013 *) -
PARI
Vec(x/((1 - x)^2*(1 - 7*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020
Formula
a(n) = (7^(n+1) - 6*n - 7)/36. - Rolf Pleisch, Oct 19 2010
a(1)=1, a(2)=9, a(3)=66; for n > 3, a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3). - Harvey P. Dale, Jul 22 2013
a(n) = Sum_{i=0..n-1} 6^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
G.f.: x/((1 - x)^2*(1 - 7*x)). - Colin Barker, Jun 03 2020
E.g.f.: exp(x)*(7*exp(6*x) - 6*x - 7)/36. - Elmo R. Oliveira, Mar 29 2025