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 A014827 #44 Mar 29 2025 17:24:43
%S A014827 1,7,38,194,975,4881,24412,122068,610349,3051755,15258786,76293942,
%T A014827 381469723,1907348629,9536743160,47683715816,238418579097,
%U A014827 1192092895503,5960464477534,29802322387690,149011611938471,745058059692377,3725290298461908,18626451492309564,93132257461547845
%N A014827 a(1)=1, a(n) = 5*a(n-1) + n.
%H A014827 Vincenzo Librandi, <a href="/A014827/b014827.txt">Table of n, a(n) for n = 1..1000</a>
%H A014827 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 pp. 9, 18.
%H A014827 László Tóth, <a href="https://arxiv.org/abs/2002.06584">On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers</a>, arXiv:2002.06584 [math.NT], 2020. See also <a href="https://doi.org/10.1090/proc/14863">Proc. Amer. Math. Soc.</a> 148 (2020), pp. 461-469.
%H A014827 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11,5).
%F A014827 a(n) = (5^(n+1) - 4*n - 5)/16.
%F A014827 G.f.: x/((1-5*x)*(1-x)^2).
%F A014827 From _Paul Barry_, Jul 30 2004: (Start)
%F A014827 a(n) = Sum_{k=0..n} (n-k)*5^k = Sum_{k=0..n} k*5^(n-k).
%F A014827 a(n) = Sum_{k=0..n} binomial(n+2,k+2)*4^k [Offset 0]. (End)
%F A014827 From _Elmo R. Oliveira_, Mar 29 2025: (Start)
%F A014827 E.g.f.: exp(x)*(5*exp(4*x) - 4*x - 5)/16.
%F A014827 a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 3. (End)
%p A014827 a:=n->sum((5^(n-j)-1^(n-j))/4,j=0..n): seq(a(n), n=1..21); # _Zerinvary Lajos_, Jan 04 2007
%t A014827 Join[{a=1,b=7},Table[c=6*b-5*a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2011 *)
%o A014827 (Sage) [(gaussian_binomial(n,1,5)-n)/4 for n in range(2,23)] # _Zerinvary Lajos_, May 29 2009
%o A014827 (Magma) [(5^(n+1)-4*n-5)/16: n in [1..30]]; // _Vincenzo Librandi_, Aug 23 2011
%Y A014827 Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016218, A016256, A023000, A023001.
%K A014827 nonn,easy
%O A014827 1,2
%A A014827 _N. J. A. Sloane_