A014831 a(1)=1; for n>1, a(n) = 8*a(n-1) + n.
1, 10, 83, 668, 5349, 42798, 342391, 2739136, 21913097, 175304786, 1402438299, 11219506404, 89756051245, 718048409974, 5744387279807, 45955098238472, 367640785907793, 2941126287262362, 23529010298098915, 188232082384791340, 1505856659078330741, 12046853272626645950
Offset: 1
Examples
For n=5, a(5) = 1*15 + 7*20 + 7^2*15 + 7^3*6 + 7^4*1 = 5349. [_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 (10,-17,8).
Programs
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Maple
a:=n->sum((8^(n-j)-1)/7,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007 a:= n-> (Matrix ([[1, 0, 1], [1, 1, 1], [0, 0, 8]])^n)[2, 3]: seq (a(n), n=1..25); # Alois P. Heinz, Aug 06 2008
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Mathematica
Table[(8^(n + 1) - 7 n - 8)/49, {n, 1, 25}] (* Bruno Berselli, Nov 13 2015 *) nxt[{n_,a_}]:={n+1,8a+n+1}; NestList[nxt,{1,1},30][[;;,2]] (* Harvey P. Dale, Aug 09 2025 *) -
PARI
Vec(x/((1 - x)^2*(1 - 8*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020
Formula
a(n) = (8^(n+1) - 7*n - 8)/49. - Rolf Pleisch, Oct 21 2010
a(n) = Sum_{i=0..n-1} 7^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
From Colin Barker, Jun 03 2020: (Start)
G.f.: x/((1 - x)^2*(1 - 8*x)).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3) for n > 3. (End)
E.g.f.: exp(x)*(8*exp(7*x) - 7*x - 8)/49. - Elmo R. Oliveira, Mar 29 2025