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A007689
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a(n) = 2^n + 3^n.
(Formerly M1444)
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91
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2, 5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697, 847322163875
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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REFERENCES
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L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 14.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015-2017.
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FORMULA
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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).
Sum_{j=0..n-1} a(j) = (1/2)*(3^n - 1) + (2^n - 1). [Jolley] - Gary W. Adamson, Dec 20 2006
Equals double binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
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
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MAPLE
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MATHEMATICA
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Table[2^n + 3^n, {n, 0, 25}]
LinearRecurrence[{5, -6}, {2, 5}, 30] (* nearly 20 times faster than the above program for large numbers. *) (* Harvey P. Dale, Oct 20 2013 *)
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PROG
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(Sage) [lucas_number2(n, 5, 6)for n in range(0, 27)] # Zerinvary Lajos, Jul 08 2008
(Haskell)
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CROSSREFS
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For odd-indexed members divided by 5 see A096951.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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