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A087455
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Expansion of (1-x)/(1-2*x+3*x^2).
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12
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1, 1, -1, -5, -7, 1, 23, 43, 17, -95, -241, -197, 329, 1249, 1511, -725, -5983, -9791, -1633, 26107, 57113, 35905, -99529, -306773, -314959, 290401, 1525679, 2180155, -216727, -6973919, -13297657, -5673557, 28545857, 74112385, 62587199, -97162757, -382087111, -472685951
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OFFSET
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0,4
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COMMENTS
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Type 2 generalized Gaussian Fibonacci integers.
Binomial transform of A077966 . [From Philippe DELEHAM, Dec 02 2008]
The real component of Q^n, where Q is the quaternion 1+0*i+1*j+1*k. - Stanislav Sykora, Jun 11 2012.
If entries are multiplicated by (-2)^n, which gives 2, -2, -2, 10, -14, -2, 46, -86, 34, 190, -482, 394,..., we obtain the Lucas V(-2,3) sequence. - R. J. Mathar, Jan 08 2013
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REFERENCES
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S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).
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LINKS
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Table of n, a(n) for n=0..37.
C. Dement, The Math Forum.
Wikipedia, Lucas sequence
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FORMULA
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Additional formulae from Paul Barry, Sep 03 2004: a(n)=2a(n-1)-3a(n-2); a(n)=(-1)^n*sum{m=0..n, binomial(n, m)*sum{k=0..n, binomial(m, 2k)2^(m-k)}}; binomial transform of 1/(1+2x^2), or (1, 0, -2, 0, 4, 0, -8, 0, 16, ...).
a(n)=(3^(n/2))cos(n*arctan(sqrt(2))).
a(n) = sqrt[ves(x^n)]/3 - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jul 31 2004
a(n+1) = a(n+2) - 2*A088137(n+1), a(n+1) = A088137(n+2) - A088137(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Oct 28 2004
a(n) = 2*a(n-1) - 3*a(n-2), n>1 a(n) = upper left and lower right terms of [1,-2, 1,1]^n. - Gary W. Adamson, Mar 28 2008
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*(-2)^(n-k). [From Philippe DELEHAM, Nov 14 2008]
a(n)=Sum_{k, 0<=k<=n}A124182(n,k)*(-3)^(n-k). [From Philippe DELEHAM, Nov 15 2008]
a(n)=(1/2)*{[1-I*sqrt(2)]^n+[1+I*sqrt(2)]^n}, with n>=0 and I=sqrt(-1) [From Paolo P. Lava, Nov 20 2008]
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MAPLE
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Digits:=100; a:=n->round(abs(evalf((3^(n/2))*cos(n*arctan(sqrt(2))))));
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PROG
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(PARI) {a(n)=if(n<0, 0, real((1+quadgen(-8))^n))} /* Michael Somos Jul 26 2006 */
(PARI) {a(n)=if(n<0, 0, subst(poltchebi(n), 'x, quadgen(12)/3)*quadgen(12)^n)} /* Michael Somos Jul 26 2006 */
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CROSSREFS
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Cf. A088137, A084102, A088138.
Cf. A048473.
Sequence in context: A108763 A061415 A196847 * A192040 A117759 A021640
Adjacent sequences: A087452 A087453 A087454 * A087456 A087457 A087458
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KEYWORD
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easy,sign
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AUTHOR
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Simone Severini, Oct 23 2003. The explicit formula was given by Paul Barry.
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EXTENSIONS
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Corrected and extended by N. J. A. Sloane, Aug 01, 2004
More terms from Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jul 31 2004
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STATUS
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approved
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