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1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Contribution from Matthew Lehman (matt.comicopia(AT)gmail.com), Nov 17 2008: (Start)
In the Fibonacci sequence, F(n) = F(n-1) + F(n-2),
for every i-th number, F(n+i) = A(i)*F(n) + B(i)*F(n-i),
B(i) is given by this sequence,
where B(i) = (-1)^(i+1).
A(i) = F(2*i-1)/F(i-1).
For every Fibonacci number, F(n+1) = F(n) + F(n-1).
For every 2nd Fibonacci number, F(n+2) = 3*F(n) - F(n-2).
For every 3rd Fibonacci number, F(n+3) = 4*F(n) + F(n-3).
For every 4th Fibonacci number, F(n+4) = 7*F(n) - F(n-4).
For every 5th Fibonacci number, F(n+5) = 11*F(n) + F(n-5).
(End) (From Vasiliy Danilov (danilovv(AT)usa.net))
Furthermore requiring F(0) = F(1) = 1, we have a(n) = (-1)^n = F(n)^2 - F(n - 1)F(n + 1), meaning that in trying to convert a square F(n)^2 square units in area into a rectangle F(n - 1) by F(n + 1) square units in area, the square will be deficient by one square unit when n is odd, and have a surplus of one square unit when n is even. (From Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 30 2009)
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=charpoly(A,2). [From Milan R. Janjic (agnus(AT)blic.net), Jan 26 2010]
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LINKS
| Tanya Khovanova, Recursive Sequences
M. Somos, Rational Function Multiplicative Coefficients
Eric Weisstein's World of Mathematics, Inverse Tangent
Eric Weisstein's World of Mathematics, Stirling Transform
Wikipedia, Grandi's series [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 21 2009]
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| G.f.: 1/(1+x). E.g.f.: exp(-x). D.g.f.: (2^(1-s)-1)*zeta(s).
Linear recurrence: a(0)=1, a(n)=-a(n-1) for n>0 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 20 2009]
Sum_{k>=0} a(k)/(k+1) = log(2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 30 2010]
Euler transform of length 2 sequence [ -1, 1]. - Michael Somos Mar 21 2011
Moebius transform is length 2 sequence [ -1, 2]. - Michael Somos Mar 21 2011
a(n) = -b(n) where b(n) = multiplicative with b(2^e) = 1 if e>1, b(p^e) = -1 if p>2 and e>1. - Michael Somos Mar 21 2011
a(n) = a(-n) = a(n + 2) = cos( n * pi). a(n) = c_2(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos Mar 21 2011
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EXAMPLE
| 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 + ...
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MAPLE
| A033999 := n->(-1)^n;
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MATHEMATICA
| Table[(-1)^n, {n, 0, 88}]
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PROG
| (PARI) a(n)=1-2*(n%2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 20 2009]
(Haskell)
a033999 n = a033999_list !! n
a033999_list = iterate (0 -) 1 -- Reinhard Zumkeller, Jan 02 2012
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CROSSREFS
| Sum_{0<=k<=n} a(k) = A059841(n) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 21 2009]
Sequence in context: A143622 A076479 A155040 * A162511 A157895 A063747
Adjacent sequences: A033996 A033997 A033998 * A034000 A034001 A034002
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KEYWORD
| sign,easy
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AUTHOR
| Vasiliy Danilov (danilovv(AT)usa.net) Jun 15 1998
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EXTENSIONS
| Comment on Fibonacci square unit creation fallacy and Mathematica command added by Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 30 2009
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