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A088138
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Generalized Gaussian Fibonacci integers.
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8
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0, 1, 2, 0, -8, -16, 0, 64, 128, 0, -512, -1024, 0, 4096, 8192, 0, -32768, -65536, 0, 262144, 524288, 0, -2097152, -4194304, 0, 16777216, 33554432, 0, -134217728, -268435456, 0, 1073741824, 2147483648, 0, -8589934592, -17179869184, 0, 68719476736, 137438953472
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The sequence 0,1,-2,0,8,-16,... has G.f. 1/(1+2x-4x^2), a(n)=2^n*sin(2n*pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)x)/sqrt(3):0,1,-3,0,9,...
a(n+1) is the Hankel transform of A100192. - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
a(n+1) is the Trinomial transform of A010892: a(n+1) = Sum[Trinomial[n,k]A010892[k+1], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907); - Paul Barry (pbarry(AT)wit.ie), Sep 10 2007
a(n+1) is the Hankel transform of A100067. [From Paul Barry (pbarry(AT)wit.ie), Jun 16 2009]
1) a(n)=A131577*A128834 2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244. 3) Sequence is identical to every 2n-th differences divided by (-3)^n. 4) a(3n)+a(3n+1)+a(3n+2)=3,-24,192,=3*A001018 signed.
5) For missing terms in a(n) see A013731=4*A001018. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 04 2009]
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (2,-4)
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FORMULA
| G.f.: 1/(1-2x+4x^2); E.g.f. : exp(x)sin(sqrt(3)x)/sqrt(3); a(n)=2a(n-1)-4a(n-2), a(0)=0, a(1)=1; a(n)=((1+isqrt(3))^n-(1-isqrt(3))^n)/(2isqrt(3)); a(n)=Im{(1+isqrt(3))^n/sqrt(3)}; a(n)=sum{k=0..floor(n/2), C(n, 2k+1)(-3)^k}.
a(n)=a(n-1)+a(n-2)+2a(n-3). - Paul Curtz, Oct 04 2009
a(n)=2a(n-1)-a(n-2)+2a(n-3). - Paul Curtz, Oct 04 2009
a(n)=a(n-1)+2a(n-2)-a(n-3)-a(n-4). - Paul Curtz, Oct 04 2009
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MATHEMATICA
| Join[{a=0, b=1}, Table[c=2*b-4*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 2, 4) for n in xrange(0, 39)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
| Cf. A084102, A088137, A045873, A088139.
Sequence in context: A009794 A171402 A104506 * A186033 A120559 A120555
Adjacent sequences: A088135 A088136 A088137 * A088139 A088140 A088141
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KEYWORD
| easy,sign
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 20 2003
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