This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014917 #60 May 24 2025 10:34:13
%S A014917 1,11,86,586,3711,22461,131836,756836,4272461,23803711,131225586,
%T A014917 717163086,3890991211,20980834961,112533569336,600814819336,
%U A014917 3194808959961,16927719116211,89406967163086,470876693725586,2473592758178711,12964010238647461,67800283432006836,353902578353881836
%N A014917 a(1)=1, a(n) = n*5^(n-1) + a(n-1).
%C A014917 From _Gary Detlefs_, Aug 31 2021 (Start)
%C A014917 This is the x=5 member of the x-family of sequences with member a(x, n) = x^n*Sum_{k=1..n} S(x, k), with S(x, k) = Sum_{j=1..k} 1/x^j.
%C A014917 S(x, k) = (x^k - 1)/((x-1)*x^k) = (1/x^k)*Sum_{j=0..k-1} x^j, and a(x,n) = ((n*(x-1) - 1)*x^n + 1)/(x-1)^2.
%C A014917 The sequence {x^k*S(x, k)} with recurrence signature (x+1, -x) leads to sequence {a(x, n)} with recurrence signature (2*x+1, -x*(x+2), x^2). (End) [Rewritten by _Wolfdieter Lang_, Nov 30 2021]
%H A014917 Vincenzo Librandi, <a href="/A014917/b014917.txt">Table of n, a(n) for n = 1..1000</a>
%H A014917 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. 16.
%H A014917 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-35,25).
%F A014917 From _Vincenzo Librandi_, Oct 23 2012: (Start)
%F A014917 a(n) = 10*a(n-1) - 25*a(n-2) + 1; a(1)=1, a(2)=11.
%F A014917 G.f.: x/((1-x)*(1-5*x)^2). (End)
%F A014917 a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3); a(1)=1, a(2)=11, a(3)=86. - _Harvey P. Dale_, May 06 2013
%F A014917 a(n) = 5^n*Sum_{k=1..n} (Sum_{j=1..k} 1/x^j) = ((4*n - 1)*5^n + 1)/4^2. See the general comment above, and the first formula. - _Gary Detlefs_, Aug 31 2021 [Edited by _Wolfdieter Lang_, Nov 30 2021]
%F A014917 E.g.f.: exp(x)*(1 + exp(4*x)*(20*x - 1))/16. - _Elmo R. Oliveira_, May 24 2025
%t A014917 CoefficientList[Series[1/((1 - x)(1 - 5 x)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 23 2012 *)
%t A014917 LinearRecurrence[{11,-35,25},{1,11,86},20] (* _Harvey P. Dale_, May 06 2013 *)
%o A014917 (Magma) I:=[1, 11]; [n le 2 select I[n] else 10*Self(n-1)-25*Self(n-2)+ 1: n in [1..30]]; // _Vincenzo Librandi_, Oct 23 2012
%Y A014917 Cf. A000351, A003463, A014915, A014916, A014917, A014918, A014920, A014921, A014923.
%K A014917 nonn,easy
%O A014917 1,2
%A A014917 _Olivier Gérard_