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 A016137 #41 Nov 14 2024 02:09:56
%S A016137 1,9,63,405,2511,15309,92583,557685,3352671,20135709,120873303,
%T A016137 725416965,4353033231,26119793709,156723545223,940355620245,
%U A016137 5642176768191,33853189749309,203119525916343,1218718317759525,7312313393341551,43873890820402509,263243376303474663,1579460351964026805,9476762394213697311
%N A016137 Expansion of 1/((1-3*x)*(1-6*x)).
%H A016137 G. C. Greubel, <a href="/A016137/b016137.txt">Table of n, a(n) for n = 0..1000</a>
%H A016137 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-18).
%F A016137 a(n) = (3^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
%F A016137 a(n) = 2*6^n - 3^n.
%F A016137 E.g.f.: (d^2/dx^2)((((exp(3*x)-1)/3)^2)/2!) = -exp(3*x) + 2*exp(6*x).
%F A016137 With leading zero, this is (6^n - 3^n)/3, the binomial transform of A016127 (with extra leading zero). - _Paul Barry_, Aug 20 2003
%F A016137 With leading zero, this is the fourth binomial transform of A001045, with a(n) = (2^n-1)(3^n/3 - 0^n/3) = A000225(n)*(A000244(n-1) - 0^n/3). - _Paul Barry_, Apr 28 2004
%F A016137 a(n) = Sum_{k=0..n} A100852(n,k). - _Reinhard Zumkeller_, Nov 20 2004
%F A016137 Sum_{k=1..n} 3^(k-1)*3^(n-k)*binomial(n, k). - _Zerinvary Lajos_, Sep 24 2006
%F A016137 a(n) = 9*a(n-1) - 18*a(n-2), n >= 2. - _Vincenzo Librandi_, Mar 14 2011
%t A016137 Table[2*6^n -3^n, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2011 *)
%t A016137 CoefficientList[Series[1/((1-3x)(1-6x)),{x,0,40}],x] (* or *) LinearRecurrence[{9,-18},{1,9},40] (* _Harvey P. Dale_, Jul 07 2012 *)
%o A016137 (Sage) [lucas_number1(n,9,18) for n in range(1,41)] # _Zerinvary Lajos_, Apr 23 2009
%o A016137 (PARI) Vec(1/(1-3*x)/(1-6*x)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%o A016137 (Magma) [2*6^n -3^n: n in [0..40]]; // _G. C. Greubel_, Nov 14 2024
%Y A016137 Second column of triangle A075498.
%Y A016137 Cf. A000225, A000244, A001045, A008277, A016127, A016129, A017933, A100852.
%K A016137 nonn,easy
%O A016137 0,2
%A A016137 _N. J. A. Sloane_
%E A016137 More terms added by _G. C. Greubel_, Nov 14 2024