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)

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

Offset

0,1

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..500

T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.

R. K. Guy, Letter to N. J. A. Sloane, Feb 1986

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

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

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

Cf. A000032, A000045, A005013, A005013, A005246, A005248.

Sequence in context: A348747 A082833 A101709 * A363550 A355715 A280104

Adjacent sequences: A005244 A005245 A005246 * A005248 A005249 A005250

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Additional comments from Michael Somos, May 01 2000

Status

approved