A014829 a(1)=1, a(n) = 6*a(n-1) + n.
1, 8, 51, 310, 1865, 11196, 67183, 403106, 2418645, 14511880, 87071291, 522427758, 3134566561, 18807399380, 112844396295, 677066377786, 4062398266733, 24374389600416, 146246337602515, 877478025615110, 5264868153690681, 31589208922144108, 189535253532864671, 1137211521197188050
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. 18.
- László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), pp. 461-469.
- Index entries for linear recurrences with constant coefficients, signature (8,-13,6).
Programs
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Magma
[(6^(n+1)-5*n-6)/25: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011
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Maple
a:=n->1/5*sum(6^j-1,j=1..n): seq(a(n),n=1..20); # Zerinvary Lajos, Jun 27 2007
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Mathematica
Join[{a=1,b=8},Table[c=7*b-6*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *) nxt[{n_,a_}]:={n+1,6a+n+1}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Feb 12 2023 *) -
PARI
Vec(x/((1 - x)^2*(1 - 6*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020
Formula
a(n) = (6^(n+1) - 5*n - 6)/25. - Rolf Pleisch, Oct 25 2010
Binomial transform of x*(1+x)/(1-5*x), or A003948 with a leading 0. a(n) = Sum_{k=0..n} (n-k)*6^k = Sum_{k=0..n} k*6^(n-k); a(n) = Sum_{k=0..n} binomial(n+2,k+2)*5^k [Offset 0]. - Paul Barry, Jul 30 2004
From Colin Barker, Jun 03 2020: (Start)
G.f.: x/((1 - x)^2*(1 - 6*x)).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3) for n > 3. (End)
E.g.f.: exp(x)*(6*exp(5*x) - 5*x - 6)/25. - Elmo R. Oliveira, Mar 29 2025