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A077925
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Expansion of 1/((1-x)*(1+2*x)).
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25
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1, -1, 3, -5, 11, -21, 43, -85, 171, -341, 683, -1365, 2731, -5461, 10923, -21845, 43691, -87381, 174763, -349525, 699051, -1398101, 2796203, -5592405, 11184811, -22369621, 44739243, -89478485, 178956971, -357913941, 715827883, -1431655765, 2863311531, -5726623061
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n+1) is the reflection of a(n) through a(n-1) on the numberline. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Aug 31 2004
If a zero is added as the (new) a(0) in front, the sequence represents the inverse binomial transform of A001045. Partial sums are in A077898. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2008]
a(n) = A077953(2*n+3). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 07 2008]
Related to the Fibonacci sequence by an INVERT transform: if A(x) = 1+x^2*g(x) is the generating function of the a(n) prefixed with 1, 0, then 1/A(x) = 2+(x+1)/(x^2-x+1) is the generating function of 1, 0, -1, 1, -2, 3,..., the signed Fibonacci sequence A000045 prefixed with 1. - Gary W. Adamson, Jan 07 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (-1,2)
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FORMULA
| G.f.: 1/(1+x-2*x^2).
a(n) = (1-(-2)^(n+1))/3. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 17 2003
a(n)=sum{k=0..n, (-2)^k } - Paul Barry (pbarry(AT)wit.ie), May 26 2003
a(n+1)-a(n)=A122803(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2008]
a(n)= Sum_{k, 0<=k<=n} A112555(n,k)*(-2)^k . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 11 2009]
a(n)= A082247(n+1)-1. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 07 2009]
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MAPLE
| a:=n->sum ((-2)^j, j=0..n): seq(a(n), n=0..35); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2008]
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MATHEMATICA
| CoefficientList[Series[(1 - x)^(-1)/(1 + 2 x), {x, 0, 50}], x] (* From Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
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PROG
| (Other) sage: [gaussian_binomial(n, 1, -2) for n in xrange(1, 35)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
(MAGMA) [(1-(-2)^(n+1))/3: n in [0..40]]; // Vincenzo Librandi, Jun 21 2011
(PARI) a(n)=(1+(-2)^n*2)/3 \\ Charles R Greathouse IV, Jun 21 2011
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CROSSREFS
| Cf. A001045 (unsigned version).
A014983, A014985, A014986 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2008]
Sequence in context: A001045 A154917 A167167 * A084230 A077465 A146574
Adjacent sequences: A077922 A077923 A077924 * A077926 A077927 A077928
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002
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