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 A083858 #41 Aug 31 2025 10:48:41
%S A083858 0,1,3,15,63,279,1215,5319,23247,101655,444447,1943271,8496495,
%T A083858 37149111,162426303,710173575,3105078543,13576277079,59359302495,
%U A083858 259535569959,1134762524847,4961500994295,21693078131967,94848240361671,414703189876815,1813199011800471
%N A083858 Expansion of x/(1 - 3*x - 6*x^2).
%C A083858 Binomial transform of A015443. A row of array A083857.
%C A083858 Pisano period lengths: 1, 1, 1, 1, 12, 1, 8, 1, 1, 12, 110, 1, 168, 8, 12, 2, 16, 1, 360, 12, ... - _R. J. Mathar_, Aug 10 2012
%H A083858 G. C. Greubel, <a href="/A083858/b083858.txt">Table of n, a(n) for n = 0..1000</a>
%H A083858 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,6).
%F A083858 a(n) = 3*a(n-1) + 6*a(n-2), a(0)=0, a(1)=1.
%F A083858 a(n) = (3*sqrt(33)/2 + 21/2)^(n/2)/sqrt(33) - (21/2 - 3*sqrt(33)/2)^(n/2)*(-1)^n/sqrt(33).
%F A083858 G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(6*k+3 + 6*x )/( x*(6*k+6 + 6*x ) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 21 2013
%F A083858 a(n) = B(n, k + 2^(n-1)) - B(n,k) where B(n,k) is formed by the family of recursions b(n) = 3*(b(n-1) + b(n-2))/2, with b(0) = 1 and b(1) = k, as explained further in A249861. - _Richard R. Forberg_, Nov 04 2014
%F A083858 a(n) = Sum_{k=0..n-1} 3^k * 2^(n-1-k) * binomial(k,n-1-k). - _Seiichi Manyama_, Aug 31 2025
%t A083858 a[n_]:=(MatrixPower[{{1,2},{1,-4}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t A083858 LinearRecurrence[{3,6}, {0,1}, 30] (* _G. C. Greubel_, Jan 16 2018 *)
%o A083858 (Sage) [lucas_number1(n,3,-6) for n in range(0, 24)] # _Zerinvary Lajos_, Apr 22 2009
%o A083858 (PARI) x='x+O('x^30); concat([0], Vec(x/(1-3*x-6*x^2))) \\ _G. C. Greubel_, Jan 16 2018
%o A083858 (Magma) I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) + 6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 16 2018
%Y A083858 Cf. A015523, A015524.
%K A083858 easy,nonn,changed
%O A083858 0,3
%A A083858 _Paul Barry_, May 06 2003