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A007689 a(n) = 2^n + 3^n.
(Formerly M1444)
85

%I M1444

%S 2,5,13,35,97,275,793,2315,6817,20195,60073,179195,535537,1602515,

%T 4799353,14381675,43112257,129271235,387682633,1162785755,3487832977,

%U 10462450355,31385253913,94151567435,282446313697,847322163875

%N a(n) = 2^n + 3^n.

%D L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 14.

%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007689/b007689.txt">Table of n, a(n) for n = 0..200</a>

%H I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, M. Stoffregen, <a href="http://arxiv.org/abs/1509.05239">Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences</a>, arXiv:1509.05239v1 [math.CO] 17 Sep 2015.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=169">Encyclopedia of Combinatorial Structures 169</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).

%F E.g.f.: exp(2*x)*(1+exp(x)). G.f.: (2-5*x)/((1-2*x)*(1-3*x)). a(n) = 5*a(n-1)-6*a(n-2).

%F 2 + 5 + 13 + 35 +...n terms = (1/2)*(3^n - 1)+(2^n - 1). [Jolley] - _Gary W. Adamson_, Dec 20 2006

%F Equals double binomial transform of [2, 1, 1, 1,...]. - _Gary W. Adamson_, Apr 23 2008

%F 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 a(n) = 2*a(n-1)+3^(n-1), with a(0)=2. - _Vincenzo Librandi_, Nov 18 2010

%F a(n) = A001550(n) - 1 = A000079(n) + A000244(n). - _Reinhard Zumkeller_, Mar 01 2012

%p A007689:=n->2^n + 3^n: seq(A007689(n), n=0..50); # _Wesley Ivan Hurt_, Jan 24 2017

%t Table[2^n + 3^n, {n, 0, 25}]

%t 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 LinearRecurrence[{5,-6},{2,5},30] (* nearly 20 times faster than the above program for large numbers. *) (* _Harvey P. Dale_, Oct 20 2013 *)

%o (Sage) [lucas_number2(n,5,6)for n in xrange(0,27)] # _Zerinvary Lajos_, Jul 08 2008

%o (PARI) a(n)=2^n+3^n \\ _Charles R Greathouse IV_, Jun 15 2011

%o (Haskell)

%o a007689 n = a000079 n + a000244 n -- _Reinhard Zumkeller_, Apr 28 2013

%Y Binomial transform of A000051.

%Y Cf. A034472, A052539, A034474, A062394, A034491, A062395, A062396, A063376, A063481, A074600-A074624.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E Additional comments from _Michael Somos_, Jun 10 2000

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Last modified May 25 05:45 EDT 2019. Contains 323539 sequences. (Running on oeis4.)