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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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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = Fibonacci(k) for k = 1, ..., n+1. - Michael Somos, Jan 08 2012
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REFERENCES
| A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
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..500
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| 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(n)=F(n) if n even, a(n)=L(n) if n odd. a(n)=F(n+1)+(-1)^(n+1)F(n-1). - Mario Catalani (mario.catalani(AT)unito.it), Sep 20 2002
G.f.: (x+x^2+x^3)/((1-x-x^2)*(1+x-x^2)).
a(n)=Product[(1 + 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]. [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)tahoo.com), Nov 26 2008]
a(2*n) = A001906(n) (Fibonacci). a(2*n + 1) = A002878(n) (Lucas). - Michael Somos, Jan 08 2012
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EXAMPLE
| 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
| with(combinat): A005013 := n-> if n mod 2 = 0 then fibonacci(n) else fibonacci(n+1)+fibonacci(n-1); fi;
A005013:=z*(z**2+z+1)/((z**2+z-1)*(z**2-z-1)); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| 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]] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)tahoo.com), Nov 26 2008]
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PROG
| (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
-- Reinhard Zumkeller, Jan 10 2012
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CROSSREFS
| Cf. A000032, A000045, A001906, A002878, A005247.
Sequence in context: A205965 A100492 A072183 * A086564 A200073 A080777
Adjacent sequences: A005010 A005011 A005012 * A005014 A005015 A005016
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Michael Somos, Jun 01 2000.
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