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 A014825 #105 Jan 23 2025 12:41:36
%S A014825 1,6,27,112,453,1818,7279,29124,116505,466030,1864131,7456536,
%T A014825 29826157,119304642,477218583,1908874348,7635497409,30541989654,
%U A014825 122167958635,488671834560,1954687338261,7818749353066
%N A014825 a(n) = 4*a(n-1) + n with n > 1, a(1)=1.
%H A014825 Vincenzo Librandi, <a href="/A014825/b014825.txt">Table of n, a(n) for n = 1..1000</a>
%H A014825 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 A014825 Hacène Belbachir and El-Mehdi Mehiri, <a href="https://arxiv.org/abs/2210.08657">Enumerating moves in the optimal solution of the Tower of Hanoi</a>, arXiv:2210.08657 [math.CO], 2022.
%H A014825 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), 461-469.
%H A014825 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,4).
%F A014825 a(n) = (4^(n+1) - 3*n - 4)/9.
%F A014825 G.f.: x/((1-4*x)*(1-x)^2).
%F A014825 a(n) = Sum_{k=0..n} (n-k)*4^k = Sum_{k=0..n} k*4^(n-k). - _Paul Barry_, Jul 30 2004
%F A014825 a(n) = Sum_{k=0..n} binomial(n+2, k+2)*3^k [Offset 0]. - _Paul Barry_, Jul 30 2004
%F A014825 a(n) = Sum_{k=0..n} Sum_{j=0..2k} (-1)^(j+1)*J(j)*J(2k-j), J(n) = A001045(n). - _Paul Barry_, Oct 23 2009
%F A014825 Convolution square of A006314. - _Michael Somos_, Jun 20 2012
%F A014825 E.g.f.: (4*exp(4*x) - (4+3*x)*exp(x))/9. - _G. C. Greubel_, Feb 18 2020
%F A014825 a(n) = A014916(-n-1)*4^(n+1) = A091919(2*n-2) for all n in Z. - _Michael Somos_, Oct 02 2020
%F A014825 a(n) = Sum_{k=0..n} A002450(k). - _Joseph Brown_, May 11 2021
%F A014825 Last digits give A171654. - _Paul Curtz_, Oct 10 2021
%e A014825 G.f. = x + 6*x^2 + 27*x^3 + 112*x^4 + 453*x^5 + 1818*x^6 + 7279*x^7 + ...
%t A014825 RecurrenceTable[{a[1]==1,a[n]==4a[n-1]+n},a[n],{n,30}] (* _Harvey P. Dale_, Oct 12 2011 *)
%t A014825 a[ n_]:= SeriesCoefficient[x/((1-4x)(1-x)^2), {x, 0, n}] (* _Michael Somos_, Jun 20 2012 *)
%o A014825 (Magma) [(4^(n+1)-3*n-4)/9: n in [1..30]]; // _Vincenzo Librandi_, Aug 23 2011
%o A014825 (PARI) {a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* _Michael Somos_, Jun 20 2012 */
%o A014825 (Sage) [(4^(n+1) -3*n -4)/9 for n in (1..30)] # _G. C. Greubel_, Feb 18 2020
%o A014825 (Python)
%o A014825 def A014825(n): return (((1<<(n+1<<1))-4)//3-n)//3 # _Chai Wah Wu_, Nov 12 2024
%Y A014825 Cf. A002450 (first differences), A052161 (partial sums).
%Y A014825 Cf. A001045, A006314, A014916, A053142, A091919.
%Y A014825 Cf. A171654 (mod 10).
%K A014825 nonn,easy
%O A014825 1,2
%A A014825 _N. J. A. Sloane_