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A014831 a(1)=1; for n>1, a(n) = 8*a(n-1) + n.

%I A014831 #29 Aug 09 2025 11:41:40
%S A014831 1,10,83,668,5349,42798,342391,2739136,21913097,175304786,1402438299,
%T A014831 11219506404,89756051245,718048409974,5744387279807,45955098238472,
%U A014831 367640785907793,2941126287262362,23529010298098915,188232082384791340,1505856659078330741,12046853272626645950
%N A014831 a(1)=1; for n>1, a(n) = 8*a(n-1) + n.
%H A014831 Colin Barker, <a href="/A014831/b014831.txt">Table of n, a(n) for n = 1..1000</a>
%H A014831 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 A014831 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,-17,8).
%F A014831 a(n) = (8^(n+1) - 7*n - 8)/49. - _Rolf Pleisch_, Oct 21 2010
%F A014831 a(n) = Sum_{i=0..n-1} 7^i*binomial(n+1,n-1-i). - _Bruno Berselli_, Nov 13 2015
%F A014831 From _Colin Barker_, Jun 03 2020: (Start)
%F A014831 G.f.: x/((1 - x)^2*(1 - 8*x)).
%F A014831 a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3) for n > 3. (End)
%F A014831 E.g.f.: exp(x)*(8*exp(7*x) - 7*x - 8)/49. - _Elmo R. Oliveira_, Mar 29 2025
%e A014831 For n=5, a(5) = 1*15 + 7*20 + 7^2*15 + 7^3*6 + 7^4*1 = 5349. [_Bruno Berselli_, Nov 13 2015]
%p A014831 a:=n->sum((8^(n-j)-1)/7,j=0..n): seq(a(n), n=1..19); # _Zerinvary Lajos_, Jan 15 2007
%p A014831 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
%t A014831 Table[(8^(n + 1) - 7 n - 8)/49, {n, 1, 25}] (* _Bruno Berselli_, Nov 13 2015 *)
%t A014831 nxt[{n_,a_}]:={n+1,8a+n+1}; NestList[nxt,{1,1},30][[;;,2]] (* _Harvey P. Dale_, Aug 09 2025 *)
%o A014831 (PARI) Vec(x/((1 - x)^2*(1 - 8*x)) + O(x^25)) \\ _Colin Barker_, Jun 03 2020
%Y A014831 Cf. A000420, A104712.
%K A014831 nonn,easy
%O A014831 1,2
%A A014831 _N. J. A. Sloane_