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 A016129 #58 Jun 01 2025 17:53:49
%S A016129 1,8,52,320,1936,11648,69952,419840,2519296,15116288,90698752,
%T A016129 544194560,3265171456,19591036928,117546237952,705277460480,
%U A016129 4231664828416,25389989101568,152339934871552,914039609753600,5484237659570176,32905425959518208,197432555761303552
%N A016129 Expansion of 1/((1-2*x)*(1-6*x)).
%H A016129 Vincenzo Librandi, <a href="/A016129/b016129.txt">Table of n, a(n) for n = 0..900</a>
%H A016129 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12).
%F A016129 a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - _Wolfdieter Lang_, Nov 07 2003
%F A016129 From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)
%F A016129 G.f.: 1/((1-2*x)*(1-6*x)).
%F A016129 E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.
%F A016129 a(n) = (6^(n+1) - 2^(n+1))/4. (End)
%F A016129 a(n)^2 = A144843(n+1). - _Philippe Deléham_, Nov 26 2008
%F A016129 a(n) = 8*a(n-1) - 12*a(n-2). - _Philippe Deléham_, Jan 01 2009
%F A016129 a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - _Mircea Merca_, Apr 06 2013
%t A016129 Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 19 2011 *)
%t A016129 CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* _Harvey P. Dale_, Jan 15 2015 *)
%o A016129 (Sage) [lucas_number1(n,8,12) for n in range(1, 31)] # _Zerinvary Lajos_, Apr 23 2009
%o A016129 (Sage) [(6^n - 2^n)/4 for n in range(1,31)] # _Zerinvary Lajos_, Jun 04 2009
%o A016129 (Magma) [(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // _Vincenzo Librandi_, Oct 09 2011
%o A016129 (PARI) Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ _Charles R Greathouse IV_, Apr 17 2012
%Y A016129 Row sums of A100851.
%Y A016129 Cf. A071951, A089278, A089500, A129467, A144843.
%Y A016129 Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).
%K A016129 nonn,easy
%O A016129 0,2
%A A016129 _N. J. A. Sloane_