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A007689
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2^n + 3^n.
(Formerly M1444)
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67
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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; internal format)
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OFFSET
| 0,1
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REFERENCES
| D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 169
Index to sequences with linear recurrences with constant coefficients, signature (5,-6).
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FORMULA
| 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).
2 + 5 + 13 + 35 +...n terms = (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 R. Janjic (agnus(AT)blic.net), May 08 2010
a(n)=2*a(n-1)+3^(n-1), with a(0)=2. - Vincenzo Librandi, Nov 18 2010
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MATHEMATICA
| Table[2^n + 3^n, {n, 0, 25}]
a=2; Numerator[Table[a=2*a-((a+1)/2), {n, 0, 7!}]](*10 times (or more) Faster! for large numbers.*) [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 19 2010]
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PROG
| sage: [lucas_number2(n, 5, 6)for n in xrange(0, 27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008
(PARI) a(n)=2^n+3^n \\ Charles R Greathouse IV, Jun 15 2011
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CROSSREFS
| Binomial transform of A000051. Cf. A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A063376, A063481, A074600 - A074624.
Sequence in context: A022855 A091190 * A085281 A082582 A086581 A059027
Adjacent sequences: A007686 A007687 A007688 * A007690 A007691 A007692
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| Additional comments from Michael Somos, Jun 10, 2000.
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