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A014292
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a(0) = a(1) = 0, a(2) = 1, a(3) = 2; for n >= 4, a(n)=2*a(n-1)-a(n-2)-a(n-4).
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3
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0, 0, 1, 2, 3, 4, 4, 2, -3, -12, -25, -40, -52, -52, -27, 38, 155, 324, 520, 678, 681, 360, -481, -2000, -4200, -6760, -8839, -8918, -4797, 6084, 25804, 54442, 87877, 115228, 116775, 63880, -76892, -332892, -705667, -1142322
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Imaginary part of the sequence of complex numbers defined by c(0) = 1, c(1) = 1, for n>1 c(n) = c(n-1) + i*c(n-2). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Apr 24 2005
a(n) = sqrt(3)*y where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,s,s,s) where s = 1/sqrt(3). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Apr 25 2005
For n>=1, a(n) is equal to -1 times the imaginary part of the determinant of the n X n matrix with the sqrt(i)'s along the superdiagonal and the subdiagonal (i is the imaginary unit), 1's along the main diagonal, and 0's everywhere else (see Mathematica code below). [John M. Campbell, June 4 2011]
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FORMULA
| a(n)=sum{k=0..floor((n+2)/2), C(n-k+2, k)sin(pi*k/2)}. - Paul Barry (pbarry(AT)wit.ie), Apr 25 2005
G.f.: x^2/(1-2x+x^2+x^4). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 22 2008]
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MATHEMATICA
| Table[-Im[Det[Array[KroneckerDelta[#1 + 1, #2]*Sqrt[I] &, {n, n}] + Array[KroneckerDelta[#1 - 1, #2]*Sqrt[I] &, {n, n}] + IdentityMatrix[n]]], {n, 1, 40}] (* From John M. Campbell, June 4 2011 *)
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CROSSREFS
| Cf. A104862, A104862.
Sequence in context: A171170 A160386 A174015 * A066078 A058339 A133852
Adjacent sequences: A014289 A014290 A014291 * A014293 A014294 A014295
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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