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A006497
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a(n) = 3a(n-1) + a(n-2).
(Formerly M0910)
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15
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2, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, 510117, 1684802, 5564523, 18378371, 60699636, 200477279, 662131473, 2186871698, 7222746567, 23855111399, 78788080764, 260219353691, 859446141837, 2838557779202
(list;
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listen;
history;
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internal format)
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OFFSET
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0,1
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COMMENTS
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Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413.
(End)
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REFERENCES
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A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.
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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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n) = [(3 + sqrt13)/2]^n + [(3 - sqrt13)/2]^n; A006190(n-2) + A006190(n) = a(n-1); [a(n)]^2 - 13[A006190(n)]^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003
E.g.f. : 2exp(3x/2)cosh(sqrt(13)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)13^k3^(n-2k)}. a(n)=2T(n, 3i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003
Comments from Hieronymus Fischer, Jan 02 2009 (Start): fract(((3+sqrt(13))/2)^n))=(1/2)*(1+(-1)^n)-(-1)^n*((3+sqrt(13))/2)^(-n)=(1/2)*(1+(-1)^n)-((3-sqrt(13))/2)^n.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which suffice x-x^(-1)=floor(x).
a(n) = round(((3+sqrt(13))/2)^n) for n>0. (End)
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 3*A097783(n), a(2n) = A057076(n).
a(3n+1) = A041018(5n), a(3n+2) = A041018(5n+3) and a(3n+3) = 2*A041018(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A006497(n) + A006190(n)*sqrt(13))/2.
Limit(A006497(n)/A006190(n), n=infinity) = sqrt(13).
(End)
G.f. : (2-3x)/(1-3x-x^2) [Dmitry Kruchinin, Feb 17 2012]
a(n) = sqrt(13*(A006190(n))^2 + 4*(-1)^n). - Vladimir Shevelev, Mar 13 2013
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MAPLE
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A006497:=(-2+3*z)/(-1+3*z+z**2); [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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Table[LucasL[n, 3], {n, 0, 26}] [Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]
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PROG
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(Sage) [lucas_number2(n, 3, -1) for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
(MAGMA) [ n eq 1 select 2 else n eq 2 select 3 else 3*Self(n-1)+Self(n-2): n in [1..40] ]; // Vincenzo Librandi, Aug 20 2011
(Haskell)
a006497 n = a006497_list !! n
a006497_list = 2 : 3 : zipWith (+) (map (* 3) $ tail a006497_list) a006497_list
-- Reinhard Zumkeller, Feb 19 2011
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CROSSREFS
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Cf. A006190.
Cf. A100230.
Cf. A001622, A014176, A080039, A098316.
Sequence in context: A159458 A057838 A219497 * A038912 A019361 A093804
Adjacent sequences: A006494 A006495 A006496 * A006498 A006499 A006500
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Ray Chandler, Feb 14 2004
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STATUS
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approved
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