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%I A007689 M1444 #83 Mar 13 2023 05:54:22
%S A007689 2,5,13,35,97,275,793,2315,6817,20195,60073,179195,535537,1602515,
%T A007689 4799353,14381675,43112257,129271235,387682633,1162785755,3487832977,
%U A007689 10462450355,31385253913,94151567435,282446313697,847322163875
%N A007689 a(n) = 2^n + 3^n.
%D A007689 L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 14.
%D A007689 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.
%D A007689 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007689 T. D. Noe, <a href="/A007689/b007689.txt">Table of n, a(n) for n = 0..200</a>
%H A007689 I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, <a href="http://arxiv.org/abs/1509.05239">Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences</a>, arXiv:1509.05239 [math.CO], 2015-2017.
%H A007689 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=169">Encyclopedia of Combinatorial Structures 169</a>
%H A007689 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F A007689 E.g.f.: exp(2*x)*(1+exp(x)).
%F A007689 G.f.: (2-5*x)/((1-2*x)*(1-3*x)).
%F A007689 a(n) = 5*a(n-1) - 6*a(n-2).
%F A007689 Sum_{j=0..n-1} a(j) = (1/2)*(3^n - 1) + (2^n - 1). [Jolley] - _Gary W. Adamson_, Dec 20 2006
%F A007689 Equals double binomial transform of [2, 1, 1, 1, ...]. - _Gary W. Adamson_, Apr 23 2008
%F A007689 If p[i] = Fibonacci(2i-5) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - _Milan Janjic_, May 08 2010
%F A007689 a(n) = 2*a(n-1) + 3^(n-1), with a(0)=2. - _Vincenzo Librandi_, Nov 18 2010
%F A007689 a(n) = A001550(n) - 1 = A000079(n) + A000244(n). - _Reinhard Zumkeller_, Mar 01 2012
%p A007689 A007689:=n->2^n + 3^n: seq(A007689(n), n=0..50); # _Wesley Ivan Hurt_, Jan 24 2017
%t A007689 Table[2^n + 3^n, {n, 0, 25}]
%t A007689 a=2;Numerator[Table[a=2*a-((a+1)/2),{n,0,7!}]] (*10 times (or more) faster for large numbers.*) (* _Vladimir Joseph Stephan Orlovsky_, Apr 19 2010 *)
%t A007689 LinearRecurrence[{5,-6},{2,5},30] (* nearly 20 times faster than the above program for large numbers. *) (* _Harvey P. Dale_, Oct 20 2013 *)
%o A007689 (Sage) [lucas_number2(n,5,6)for n in range(0,27)] # _Zerinvary Lajos_, Jul 08 2008
%o A007689 (PARI) a(n)=2^n+3^n \\ _Charles R Greathouse IV_, Jun 15 2011
%o A007689 (Haskell)
%o A007689 a007689 n = a000079 n + a000244 n  -- _Reinhard Zumkeller_, Apr 28 2013
%o A007689 (Magma) [2^n+3^n: n in [0..30]]; // _G. C. Greubel_, Mar 11 2023
%Y A007689 For odd-indexed members divided by 5 see A096951.
%Y A007689 Binomial transform of A000051.
%Y A007689 Cf. A000079, A000244, A001550, A063376, A063481.
%Y A007689 Cf. A074600 - A074624, A082101 (primes).
%K A007689 nonn,easy,nice
%O A007689 0,1
%A A007689 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A007689 Additional comments from _Michael Somos_, Jun 10 2000

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