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A006495
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Real part of (1 + 2*i)^n, where i is sqrt(-1).
(Formerly M2880)
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22
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1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 237, 6469, 11753, -8839, -76443, -108691, 164833, 873121, 922077, -2521451, -9653287, -6699319, 34867797, 103232189, 32125393, -451910159, -1064447283, 130656229, 5583548873
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OFFSET
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0,3
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COMMENTS
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Binomial transform of [1, 0, -4, 0, 16, 0, -64, 0, 256, 0, ...], i.e. powers of -4 with interpolated zeros. - Philippe Deléham, Dec 02 2008
The absolute values of these numbers are the odd numbers y such that x^2 + y^2 = 5^n with x and y coprime. See A098122. - T. D. Noe, Apr 14 2011
Pisano period lengths: 1, 1, 8, 1, 4, 8, 48, 4, 24, 4, 60, 8, 12, 48, 8, 8, 16, 24, 90, 4, ... - R. J. Mathar, Aug 10 2012
Multiplied by a signed sequence of 2's we obtain 2, -2, -6, 22, -14, -82, 234, -58, -1054, 2398, 474, -12938, ..., the Lucas V(-2,5) sequence. - R. J. Mathar, Jan 08 2013
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (1-x)/(1 - 2*x + 5*x^2);
a(n) = 2*a(n-1) - 5*a(n-2);
a(n) = 5^(n/2)*cos(n*atan(1/3) + Pi*n/4);
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,k-j)*C(j,n-k)}*(-4)^(n-k). (End)
A000351(n) = a(n)^2 + A006496(n)^2. - Fabrice Baubet (intih(AT)free.fr), May 28 2007
a(n) = upper left and lower right terms of the 2 X 2 matrix [1,-2; 2,1]^n. - Gary W. Adamson, Mar 28 2008
a(n) = (4*n+5)*a(n-1) - 8*Sum_{k=1..n} a(k-1)*a(n-k) if n > 0. - Michael Somos, Jul 23 2011
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k+1)/(x*(4*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
G.f.: Sum_{n>=0} (1 + (-1)^n*i)^n * x^n / (1 - (-1)^n*i*x)^(n+1).
G.f.: Sum_{n>=0} (1 - (-1)^n*i)^n * x^n / (1 + (-1)^n*i*x)^(n+1).
(End)
a(n) = hypergeom([1/2 - n/2, -n/2], [1/2], -4). - Peter Luschny, Jul 26 2020
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EXAMPLE
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1 + x - 3*x^2 - 11*x^3 - 7*x^4 + 41*x^5 + 117*x^6 + 29*x^7 - 527*x^8 + ...
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MAPLE
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a := n -> hypergeom([1/2 - n/2, -n/2], [1/2], -4):
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MATHEMATICA
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PROG
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(Sage) [lucas_number2(n, 2, 5)/2 for n in range(0, 30)] # Zerinvary Lajos, Jul 08 2008
(PARI) {a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (4*k + 1) * A[k-1] - 8 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */
(PARI) {a(n) = my(A=1);
A = sum(m=0, n+1, (1 + (-1)^m*I)^m * x^m / (1 - (-1)^m*I*x +x*O(x^n))^(m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Mar 09 2019
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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