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A014916 a(1)=1, a(n) = n*4^(n-1) + a(n-1).

%I A014916 #40 Jan 31 2025 14:56:26
%S A014916 1,9,57,313,1593,7737,36409,167481,757305,3378745,14913081,65244729,
%T A014916 283348537,1222872633,5249404473,22429273657,95443717689,404681363001,
%U A014916 1710351420985,7207909559865,30297653743161,127054676987449,531674956009017,2220524816272953,9257399234039353
%N A014916 a(1)=1, a(n) = n*4^(n-1) + a(n-1).
%H A014916 Vincenzo Librandi, <a href="/A014916/b014916.txt">Table of n, a(n) for n = 1..1000</a>
%H A014916 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. 15.
%H A014916 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-24,16).
%F A014916 a(1)=1, a(2)=9, a(n) = 8*a(n-1) - 16*a(n-2) + 1. - _Vincenzo Librandi_, Oct 23 2012
%F A014916 G.f.: x/((1-x)*(1-4*x)^2). - _Vincenzo Librandi_, Oct 23 2012
%F A014916 a(n) = (4^n*(3*n-1) + 1)/9. - _Thomas M. Cowley_, Jan 25 2025
%F A014916 E.g.f.: exp(x)*(1 + exp(3*x)*(12*x - 1))/9. - _Stefano Spezia_, Jan 31 2025
%p A014916 a:=n->sum (4^n-4^j, j=0..n): seq(a(n)/3, n=1..31); # _Zerinvary Lajos_, Dec 14 2008
%t A014916 Join[{a=1,b=9},Table[c=8*b-16*a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2011 *)
%t A014916 CoefficientList[Series[1/((1 - x)(1 - 4*x)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 23 2012 *)
%t A014916 LinearRecurrence[{9,-24,16},{1,9,57},30] (* _Harvey P. Dale_, Jul 25 2015 *)
%o A014916 (Magma) I:=[1, 9]; [n le 2 select I[n] else 8*Self(n-1)-16*Self(n-2)+ 1: n in [1..30]]; // _Vincenzo Librandi_, Oct 23 2012
%K A014916 nonn,easy
%O A014916 1,2
%A A014916 _Olivier Gérard_