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A108520
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Expansion of 1/(1+2*x+2*x^2).
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12
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1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, 128, -128, 0, 256, -512, 512, 0, -1024, 2048, -2048, 0, 4096, -8192, 8192, 0, -16384, 32768, -32768, 0, 65536, -131072, 131072, 0, -262144, 524288, -524288, 0, 1048576, -2097152, 2097152, 0, -4194304, 8388608, -8388608
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Yet another variation on A009545.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (-2,-2).
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FORMULA
| G.f.: 1/(1+2*x+2*x^2).
E.g.f.: exp(-x)*(cos(x)-sin(x)).
a(n)=-2*(a(n-1)+a(n-2)).
a(n)=sum{k=0..n, sum{j=0..n-k, C(k,j)C(k,n-j)(-2)^(n-j)}} - Paul Barry (pbarry(AT)wit.ie), Mar 09 2006
a(n) = -4 * a(n-4). - Paul Curtz, Apr 24 2011
a(n) = A016116(n+1) * A075553(n+1). - Paul Curtz, Apr 25 2011
a(n) = -(-1-i)^(n-1)-(-1+i)^(n-1). a(n) = -2*A009116(n-1) for n>0 - Bruno Berselli, Apr 26 2011
Imaginary part of (-1+i)^n, negated real part is A090132. [Joerg Arndt, May 13 2011]
E.g.f.: (cos(x)-sin(x))*exp(-x)=G(0); G(k)=1-2*x/(4*k+1+x*(4*k+1)/(2*(2*k+1)-x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(1/(1+2*x+2*x^2)+x*O(x^n), n))
(PARI) a(n)=if(n<1, n==0, -polsym(2+2*x+x^2, n-1)[n])
(MAGMA) [n le 2 select n*(-1)^(n-1) else -2*(Self(n-1)+Self(n-2)): n in [1..47]]; // Bruno Berselli, Apr 26 2011
(PARI) vector(66, n, imag((-1+I)^n)) /* Joerg Arndt, May 13 2011 */
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CROSSREFS
| Cf. a(n)=(-1)^n A099087(n). a(n)=-A084102(n) if n>0.
Sequence in context: A199454 A180813 A194656 * A099087 A009545 A084102
Adjacent sequences: A108517 A108518 A108519 * A108521 A108522 A108523
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KEYWORD
| sign,easy
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AUTHOR
| Michael Somos, Jun 07 2005
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