A014915 a(1)=1, a(n) = n*3^(n-1) + a(n-1).
1, 7, 34, 142, 547, 2005, 7108, 24604, 83653, 280483, 930022, 3055786, 9964519, 32285041, 104029576, 333612088, 1065406345, 3389929279, 10750918570, 33996147910, 107218620331, 337346390797, 1059110761804, 3318547053652, 10379285465677, 32408789311195, 101039166676078
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..600
- 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. 14.
- Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, arXiv:1809.00122 [math.CA], 2018.
- Index entries for linear recurrences with constant coefficients, signature (7,-15,9).
Programs
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Magma
[((2*n - 1)*3^n + 1)/4: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011
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Mathematica
LinearRecurrence[{7, -15, 9}, {1, 7, 34}, 25] (* L. Edson Jeffery, May 08 2015 *)
Formula
From Henry Bottomley, Dec 18 2000: (Start)
a(n) = ((2*n-1)*3^n + 1)/4.
a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3) for n > 3.
a(n) = 1 + 2*3 + 3*3^2 + .. + n*3^(n-1).
a(n) = a(n-1) + A027471(n+1). (End)
G.f.: x/((1-x)*(1-3*x)^2). - Colin Barker, Jul 28 2012
a(n) = f^n(n)/2 with f(x) = 3*x-1. - Glen Gilchrist, Apr 10 2019
E.g.f.: exp(x)*(1 + exp(2*x)*(6*x - 1))/4. - Stefano Spezia, May 14 2024
a(n) = 6*a(n-1) - 9*a(n-2) + 1 for n > 2. - Elmo R. Oliveira, May 24 2025