A014917 a(1)=1, a(n) = n*5^(n-1) + a(n-1).
1, 11, 86, 586, 3711, 22461, 131836, 756836, 4272461, 23803711, 131225586, 717163086, 3890991211, 20980834961, 112533569336, 600814819336, 3194808959961, 16927719116211, 89406967163086, 470876693725586, 2473592758178711, 12964010238647461, 67800283432006836, 353902578353881836
Offset: 1
Links
- Vincenzo Librandi, 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. 16.
- Index entries for linear recurrences with constant coefficients, signature (11,-35,25).
Programs
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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
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Mathematica
CoefficientList[Series[1/((1 - x)(1 - 5 x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 23 2012 *) LinearRecurrence[{11,-35,25},{1,11,86},20] (* Harvey P. Dale, May 06 2013 *)
Formula
From Vincenzo Librandi, Oct 23 2012: (Start)
a(n) = 10*a(n-1) - 25*a(n-2) + 1; a(1)=1, a(2)=11.
G.f.: x/((1-x)*(1-5*x)^2). (End)
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
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]
E.g.f.: exp(x)*(1 + exp(4*x)*(20*x - 1))/16. - Elmo R. Oliveira, May 24 2025
Comments