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A005013
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a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=1, a(3)=4. Alternates Fibonacci (A000045) and Lucas (A000032) sequences for even and odd n.
(Formerly M3214)
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17
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0, 1, 1, 4, 3, 11, 8, 29, 21, 76, 55, 199, 144, 521, 377, 1364, 987, 3571, 2584, 9349, 6765, 24476, 17711, 64079, 46368, 167761, 121393, 439204, 317811, 1149851, 832040, 3010349, 2178309, 7881196, 5702887, 20633239, 14930352, 54018521, 39088169, 141422324
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OFFSET
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0,4
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COMMENTS
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S(n,sqrt(5)), with the Chebyshev polynomials A049310, is an integer sequence in the real quadratic number field Q(sqrt(5)) with basis numbers <1,phi>, phi:=(1+sqrt(5))/2. S(n,sqrt(5)) = A(n) + 2*B(n)*phi, with A(n)= a(n+1)*(-1)^n and B(n)= A147600(n-1), n>=0, with A147600(-1):=0.
a(n) = p(n+1) where p(x) is the unique degree-(n-1) polynomial such that p(k) = Fibonacci(k) for k = 1, ..., n. - Michael Somos, Jan 08 2012
This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 5 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence satisfies a linear recurrence of order four. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -5, 1, -5, ...] = 1/(1 - 1/(5 - 1/(1 - 1/(5 - ...)))) = (1/2)*(5 - sqrt(5)) begins [0/1, 1/1, 5/4, 4/3, 15/11, 11/8, 40/29, ...]; the denominators give the present sequence. The sequence of numerators [0, 1, 5, 4, 15, 11, 40, ...] is A203976. Cf. A108412 and A026741. - Peter Bala, May 19 2014
Define a binary operation o on the real numbers by x o y = x*sqrt(1 + y^2) + y*sqrt(1 + x^2). The operation o is commutative and associative with identity 0. We have (1/2)*a(2*n + 1) = 1/2 o 1/2 o ... o 1/2 (2*n + 1 terms) and (1/2)*sqrt(5)* a(2*n) = 1/2 o 1/2 o ... o 1/2 (2*n terms). Cf. A084068 and A049629. - Peter Bala, Mar 23 2018
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REFERENCES
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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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FORMULA
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a(1) = a(2) = 1, a(3) = 4, a(n) = (a(n-1) * a(n-2) - 1) / a(n-3), unless n=3. a(-n) = -a(n).
a(2n) = A001906(n), a(2n+1) = A002878(n). a(n)=F(n+1)+(-1)^(n+1)F(n-1). - Mario Catalani (mario.catalani(AT)unito.it), Sep 20 2002
G.f.: x*(1+x+x^2)/((1-x-x^2)*(1+x-x^2)).
a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = (1/2)*(sqrt(5) + 1) and beta = (1/2)*(sqrt(5) - 1). Equivalently, a(n) = U(n-1, sqrt(5)/2) for n odd and a(n) = (1/sqrt(5))*U(n-1, sqrt(5)/2) for n even, where U(n,x) is the Chebyshev polynomial of the second kind. (End)
E.g.f.: (Phi/sqrt(5))*exp(-Phi*x)*(exp(x)-1)*(exp(sqrt(5)*x) - 1/(Phi)^2), where Phi = (1+sqrt(5))/2. - G. C. Greubel, Feb 08 2016
a(n) = (5^floor((n-1)/2)/2^(n-1))*Sum_{k=0..n-1} binomial(n-1,k)/5^floor(k/2). - Tony Foster III, Oct 21 2018
a(n) = hypergeom([(1 - n)/2, (n + 1) mod 2 - n/2], [1 - n], -4) for n >= 2. - Peter Luschny, Sep 03 2019
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EXAMPLE
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G.f. = x + x^2 + 4*x^3 + 3*x^4 + 11*x^5 + 8*x^6 + 29*x^7 + 21*x^8 + 76*x^9 + ...
a(3) = 4 since p(x) = (x^2 - 3*x + 4) / 2 interpolates p(1) = 1, p(2) = 1, p(3) = 2, and p(4) = 4. - Michael Somos, Jan 08 2012
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MAPLE
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with(combinat): A005013 := n-> if n mod 2 = 0 then fibonacci(n) else fibonacci(n+1)+fibonacci(n-1); fi;
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MATHEMATICA
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CoefficientList[Series[(x + x^2 + x^3)/(1 - 3x^2 + x^4), {x, 0, 40}], x]
f[n_] = Product[(1 + 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; a = Table[f[n], {n, 0, 30}]; Round[a]; FullSimplify[ExpandAll[a]] (* Roger L. Bagula and Gary W. Adamson, Nov 26 2008 *)
LinearRecurrence[{0, 3, 0, -1}, {0, 1, 1, 4}, 100] (* G. C. Greubel, Feb 08 2016 *)
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PROG
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(PARI) {a(n) = if( n%2, fibonacci(n+1) + fibonacci(n-1), fibonacci(n))}; /* Michael Somos, Jan 08 2012 */
(PARI) {a(n) = if( n<0, -a(-n), subst( polinterpolate( vector( n, k, fibonacci(k))), x, n+1))}; /* Michael Somos, Jan 08 2012 */
(Haskell)
a005013 n = a005013_list !! n
a005013_list = alt a000045_list a000032_list where
alt (f:_:fs) (_:l:ls) = f : l : alt fs ls
(Magma) I:=[0, 1, 1, 4]; [n le 4 select I[n] else 3*Self(n-2) - Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 09 2016
(GAP) a:=[0, 1, 1, 4];; for n in [5..40] do a[n]:=3*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Oct 21 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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