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A052952
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Fibonacci(n+2) - (1-(-1)^n)/2.
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28
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1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169
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OFFSET
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0,3
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COMMENTS
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Equals row sums of triangle A173284 [From Gary W. Adamson, Feb 14 2010]
The Kn21 sums, see A180662 for the definition of these sums, of the ‘Races with Ties’ triangle A035317 equal this sequence. [Johannes W. Meijer, Jul 20 2011]
a(n+1) = a(n) + a(n-1) + A059841(n+1). [Reinhard Zumkeller, Jan 06 2012]
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023
Index to sequences with linear recurrences with constant coefficients, signature (1,2,-1,-1).
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FORMULA
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G.f.: 1/((-1+x+x^2)*(-1+x^2)).
a(n) = A074331(n+1).
Recurrence: a(0)=1, a(1)=1, a(2)=3, a(n) = 2*a(n-2) + a(n-3) + 1.
Sum(1/5*(3+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+_Z^2))+Sum(-1/2*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))
a(2*k)=sum(F(2*j+1), j=0..k)= F(2*(k+1)), k >= 0; a(2*k-1)=sum(F(2*j), j=0..k)= F(2*k+1)-1, k >= 1; F(n)=A000045(n) (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) + (1+(-1)^n)/2.
a(n) = sum{k=0..floor(n/2), binomial(n-k+1, k)} - Paul Barry, Oct 23 2004
a(n) = floor(phi^(n+2) / sqrt(5)), where phi is the golden ratio: phi = (1+sqrt(5))/2. - Reinhard Zumkeller, Apr 19 2005
a(n) = fibonacci(n-1)+a(n-2), a(2)=1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 17 2008
Another recurrence (with a different offset) is given in the Maple code.
a(n) = floor(fibonacci(n+3)^2/fibonacci(n+4))[From Gary Detlefs Nov 29 2010]
a(n) = (A001595(n+3)-A066983(n+4))/2 [From Gary Detlefs Dec 19 2010]
a(4*n) = F(4*n+2); a(4*n+1) = F(4*n+3) - 1; a(4*n+2) = F(4*n+4); a(4*n+3) = F(4*n+5) - 1; [Johannes W. Meijer, Jul 20 2011]
a(n) = floor(|F((1+I)*(n+2))|), n>=0, with the complex Fibonacci function F: C -> C, z -> F(z) with
F(z) := (exp(log(phi)*z) - exp(I*Pi*z)*exp(-log(phi)*z))/(2*phi-1) with the modulus |z|, the imaginary unit I and the golden section phi:=(1+sqrt(5))/2. A Conjecture. For F(z) see, e.g., the T. Koshy reference. ch. 45, p. 523, where F is called f, given in A000045. [From Wolfdieter Lang, Jul 24 2012]
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MAPLE
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spec := [S, {S=Prod(Sequence(Union(Prod(Z, Z), Z)), Sequence(Prod(Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
with (combinat):a[2]:=1:for n from 2 to 50 do a[n]:=fibonacci(n-1)+a[n-2] od: seq(a[n-1], n=3..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 17 2008
f:=proc(n) option remember; local t1; if n <= 2 then RETURN(1); fi: if n mod 2 = 1 then t1:=1 else t1:=0; fi: f(n-1)+f(n-2)+t1; end; [seq(f(n), n=1..100)]; - N. J. A. Sloane, May 25 2008
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MATHEMATICA
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a=0; b=0; lst={a, b}; Do[z=a+b+1; AppendTo[lst, z]; a=b; b=z; z=a+b; AppendTo[lst, z]; a=b; b=z, {n, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Feb 16 2010]
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PROG
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(PARI) a(n)=if(n<0, 0, fibonacci(n+2)-n%2)
(Haskell)
a052952 n = a052952_list !! n
a052952_list = 1 : 1 : zipWith (+)
a059841_list (zipWith (+) a052952_list $ tail a052952_list)
-- Reinhard Zumkeller, Jan 06 2012
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CROSSREFS
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a(n)= A054450(n+1, 1) (second column of triangle).
Cf. A062114.
Cf. A173284 [From Gary W. Adamson, Feb 14 2010]
Cf. a059841, A014217.
Sequence in context: A147622 A173534 * A074331 A153339 A033955 A049720
Adjacent sequences: A052949 A052950 A052951 * A052953 A052954 A052955
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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Additional formulae and more terms from Wolfdieter Lang, May 02 2000.
Better description from Olivier Gérard, Jun 05 2001.
More terms from James A. Sellers, Jun 06 2000
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STATUS
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approved
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