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A005247 a(n) = 3*a(n-2) - a(n-4), a(0)=2, a(1)=1, a(2)=3, a(3)=2. Alternates Lucas (A000032) and Fibonacci (A000045) sequences for even and odd n.
(Formerly M0149)
6
2, 1, 3, 2, 7, 5, 18, 13, 47, 34, 123, 89, 322, 233, 843, 610, 2207, 1597, 5778, 4181, 15127, 10946, 39603, 28657, 103682, 75025, 271443, 196418, 710647, 514229, 1860498, 1346269, 4870847, 3524578, 12752043, 9227465, 33385282, 24157817 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
a(0)=2, a(1)=1, a(2)=3, a(n) = (1+a(n-1)a(n-2))/a(n-3), n >= 3. a(-n) = a(n).
G.f.: (2+x-3*x^2-x^3)/((1-x-x^2)*(1+x-x^2))
a(n) = F(n) if n odd, a(n) = L(n) if n even. a(n) = F(n+1)+(-1)^nF(n-1). - Mario Catalani (mario.catalani(AT)unito.it), Sep 20 2002
a(n) = ((5+sqrt(5))/10)*(((1+sqrt(5))/2)^n+((-1+sqrt(5))/2)^n)+((5-sqrt(5))/10)*(((1-sqrt(5))/2)^n+((-1-sqrt(5))/2)^n). With additional leading 1: a(n)=((sqrt(5))/5)*(((1+sqrt(5))/2)^n-((1-sqrt(5))/2)^n)+((5+3*sqrt(5))/10)*((-1+sqrt(5))/2)^n+((5-3*sqrt(5))/10)*((-1-sqrt(5))/2)^n. - Tim Monahan, Jul 25 2011
From Peter Bala, Jan 11 2013: (Start)
Let phi = 1/2*(sqrt(5) - 1). This sequence is the simple continued fraction expansion of the real number 1 + product {n >= 0} (1 + sqrt(5)*phi^(4*n+1))/(1 + sqrt(5)*phi^(4*n+3)) = 2.77616 23282 02325 23857 ... = 2 + 1/(1 + 1/(3 + 1/(2 + 1/(7 + ...)))). Cf. A005248.
Furthermore, for k = 0,1,2,... the simple continued fraction expansion of 1 + product {n >= 0} (1 + 1/5^k*sqrt(5)*phi^(4*n+1))/(1 + 1/5^k*sqrt(5)*phi^(4*n+3)) equals [2; 1*5^k, 3, 2*5^k, 7, 5*5^k, 18, 13*5^k, 47, ...]. (End)
a(n) = hypergeom([(1-n)/2, n mod 2 - n/2], [1 - n], -4) for n > 2. - Peter Luschny, Sep 03 2019
E.g.f.: 2*cosh(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Mar 15 2022
MAPLE
with(combinat): A005247 := n-> if n mod 2 = 1 then fibonacci(n) else fibonacci(n+1)+fibonacci(n-1); fi;
A005247:=-(z+1)*(3*z**2-z-1)/(z**2-z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading 1.
MATHEMATICA
CoefficientList[Series[(2 + x - 3x^2 - x^3)/(1 - 3x^2 + x^4), {x, 0, 40}], x]
LinearRecurrence[{0, 3, 0, -1}, {2, 1, 3, 2}, 50] (* Harvey P. Dale, Oct 10 2012 *)
PROG
(PARI) a(n)=if(n%2, fibonacci(n), fibonacci(n+1)+fibonacci(n-1))
(Haskell)
a005247 n = a005247_list !! n
a005247_list = f a000032_list a000045_list where
f (x:_:xs) (_:y:ys) = x : y : f xs ys
-- Reinhard Zumkeller, Dec 27 2012
(Magma) I:=[2, 1, 3, 2]; [n le 4 select I[n] else 3*Self(n-2) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017
CROSSREFS
Sequence in context: A348747 A082833 A101709 * A363550 A355715 A280104
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Additional comments from Michael Somos, May 01 2000
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)