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A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).

%I A006495 M2880 #100 Jan 05 2025 19:51:34
%S A006495 1,1,-3,-11,-7,41,117,29,-527,-1199,237,6469,11753,-8839,-76443,
%T A006495 -108691,164833,873121,922077,-2521451,-9653287,-6699319,34867797,
%U A006495 103232189,32125393,-451910159,-1064447283,130656229,5583548873
%N A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).
%C A006495 Row sums of the Euler related triangle A117411. Partial sums are A006495. - _Paul Barry_, Mar 16 2006
%C A006495 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
%C A006495 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
%C A006495 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
%C A006495 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
%D A006495 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006495 Vincenzo Librandi, <a href="/A006495/b006495.txt">Table of n, a(n) for n = 0..200</a> (a(88) onwards corrected by Sean A. Irvine, Apr 29 2019)
%H A006495 Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, <a href="https://doi.org/10.1007/s00006-019-0969-9">On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis</a>, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
%H A006495 G. Berzsenyi, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-3/berzsenyi.pdf">Gaussian Fibonacci numbers</a>, Fib. Quart., 15 (1977), 233-236.
%H A006495 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>.
%H A006495 <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>.
%H A006495 <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>.
%H A006495 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-5).
%F A006495 a(n) = (1/2)*((1+2*i)^n + (1-2*i)^n). - _Benoit Cloitre_, Oct 28 2002
%F A006495 From _Paul Barry_, Mar 16 2006: (Start)
%F A006495 G.f.: (1-x)/(1 - 2*x + 5*x^2);
%F A006495 a(n) = 2*a(n-1) - 5*a(n-2);
%F A006495 a(n) = 5^(n/2)*cos(n*atan(1/3) + Pi*n/4);
%F A006495 a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,k-j)*C(j,n-k)*(-4)^(n-k). (End)
%F A006495 A000351(n) = a(n)^2 + A006496(n)^2. - Fabrice Baubet (intih(AT)free.fr), May 28 2007
%F A006495 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
%F A006495 a(n) = Sum_{k=0..n} A124182(n,k)*(-5)^(n-k). - _Philippe Deléham_, Nov 01 2008
%F A006495 a(n) = Sum_{k=0..n} A098158(n,k)*(-4)^(n-k). - _Philippe Deléham_, Nov 14 2008
%F A006495 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
%F A006495 E.g.f.: exp(x)*cos(2*x). - _Sergei N. Gladkovskii_, Jul 22 2012
%F A006495 a(n) = 5^(n/2) * cos(n*arctan(2)). - _Sergei N. Gladkovskii_, Aug 13 2012
%F A006495 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
%F A006495 From _Paul D. Hanna_, Mar 09 2019: (Start)
%F A006495 G.f.: Sum_{n>=0} (1 + (-1)^n*i)^n * x^n / (1 - (-1)^n*i*x)^(n+1).
%F A006495 G.f.: Sum_{n>=0} (1 - (-1)^n*i)^n * x^n / (1 + (-1)^n*i*x)^(n+1).
%F A006495 (End)
%F A006495 a(n) = hypergeom([1/2 - n/2, -n/2], [1/2], -4). - _Peter Luschny_, Jul 26 2020
%e A006495 1 + x - 3*x^2 - 11*x^3 - 7*x^4 + 41*x^5 + 117*x^6 + 29*x^7 - 527*x^8 + ...
%p A006495 a := n -> hypergeom([1/2 - n/2, -n/2], [1/2], -4):
%p A006495 seq(simplify(a(n)), n=0..28); # _Peter Luschny_, Jul 26 2020
%t A006495 Table[Re[(1+2I)^n],{n,0,29}] (* _Giovanni Resta_, Mar 28 2006 *)
%o A006495 (Sage) [lucas_number2(n,2,5)/2 for n in range(0,30)] # _Zerinvary Lajos_, Jul 08 2008
%o A006495 (Magma) A006495:=func< n | Integers()!Real((1+2*Sqrt(-1))^n) >; [ A006495(n): n in [0..30] ]; // _Klaus Brockhaus_, Feb 04 2011
%o A006495 (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 */
%o A006495 (PARI) a(n) = real( (1 + 2*I)^n ) \\ _Charles R Greathouse IV_, Nov 21 2014
%o A006495 (PARI) {a(n) = my(A=1);
%o A006495 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
%Y A006495 Cf. A006496, A045873 (partial sums).
%K A006495 sign,easy
%O A006495 0,3
%A A006495 _N. J. A. Sloane_
%E A006495 Signs from _Christian G. Bower_, Nov 15 1998
%E A006495 Corrected by _Giovanni Resta_, Mar 28 2006