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A014830 a(1)=1; for n > 1, a(n) = 7*a(n-1) + n.

%I A014830 #37 Mar 31 2025 14:44:30
%S A014830 1,9,66,466,3267,22875,160132,1120932,7846533,54925741,384480198,
%T A014830 2691361398,18839529799,131876708607,923136960264,6461958721864,
%U A014830 45233711053065,316635977371473,2216451841600330,15515162891202330,108606140238416331,760242981668914339,5321700871682400396
%N A014830 a(1)=1; for n > 1, a(n) = 7*a(n-1) + n.
%H A014830 Colin Barker, <a href="/A014830/b014830.txt">Table of n, a(n) for n = 1..1000</a>
%H A014830 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 A014830 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-15,7).
%F A014830 a(n) = (7^(n+1) - 6*n - 7)/36. - _Rolf Pleisch_, Oct 19 2010
%F A014830 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
%F A014830 a(n) = Sum_{i=0..n-1} 6^i*binomial(n+1,n-1-i). - _Bruno Berselli_, Nov 13 2015
%F A014830 G.f.: x/((1 - x)^2*(1 - 7*x)). - _Colin Barker_, Jun 03 2020
%F A014830 E.g.f.: exp(x)*(7*exp(6*x) - 6*x - 7)/36. - _Elmo R. Oliveira_, Mar 29 2025
%e A014830 For n=5, a(5) = 1*15 + 6*20 + 6^2*15 + 6^3*6 + 6^4*1 = 3267. - _Bruno Berselli_, Nov 13 2015
%p A014830 a:=n->sum((7^(n-j)-1)/6,j=0..n): seq(a(n), n=1..19); # _Zerinvary Lajos_, Jan 15 2007
%t A014830 a[1] = 1; a[n_] := 7*a[n-1]+n; Table[a[n], {n, 10}] (* _Zak Seidov_, Feb 06 2011 *)
%t A014830 LinearRecurrence[{9, -15, 7}, {1, 9, 66}, 30] (* _Harvey P. Dale_, Jul 22 2013 *)
%o A014830 (PARI) Vec(x/((1 - x)^2*(1 - 7*x)) + O(x^25)) \\ _Colin Barker_, Jun 03 2020
%Y A014830 Row n=7 of A126885.
%Y A014830 Cf. A000400, A104712.
%K A014830 nonn,easy
%O A014830 1,2
%A A014830 _N. J. A. Sloane_