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A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).
(Formerly M2880)
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums of the Euler related triangle A117411. Partial sums are A006495. - Paul Barry, Mar 16 2006

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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200 (a(88) onwards corrected by Sean A. Irvine, Apr 29 2019)

Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

G. Berzsenyi, Gaussian Fibonacci numbers, Fib. Quart., 15 (1977), 233-236.

Wikipedia, Lucas sequence

Index entries for Lucas sequences

Index entries for Gaussian integers and primes

Index entries for linear recurrences with constant coefficients, signature (2,-5).

FORMULA

a(n) = (1/2)*((1+2*i)^n + (1-2*i)^n). - Benoit Cloitre, Oct 28 2002

From Paul Barry, Mar 16 2006: (Start)

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) = Sum_{k=0..n} A124182(n,k)*(-5)^(n-k). - Philippe Deléham, Nov 01 2008

a(n) = Sum_{k=0..n} A098158(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 14 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

E.g.f.: exp(x)*cos(2*x). - Sergei N. Gladkovskii, Jul 22 2012

a(n) = 5^(n/2) * cos(n*arctan(2)). - Sergei N. Gladkovskii, Aug 13 2012

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

From Paul D. Hanna, Mar 09 2019: (Start)

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)

EXAMPLE

1 + x - 3*x^2 - 11*x^3 - 7*x^4 + 41*x^5 + 117*x^6 + 29*x^7 - 527*x^8 + ...

MATHEMATICA

Table[Re[(1+2I)^n], {n, 0, 29}] (* Giovanni Resta, Mar 28 2006 *)

PROG

(Sage) [lucas_number2(n, 2, 5)/2 for n in xrange(0, 30)] # Zerinvary Lajos, Jul 08 2008

(MAGMA) A006495:=func< n | Integers()!Real((1+2*Sqrt(-1))^n) >; [ A006495(n): n in [0..30] ]; // Klaus Brockhaus, Feb 04 2011

(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) = real( (1 + 2*I)^n ) \\ Charles R Greathouse IV, Nov 21 2014

(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

CROSSREFS

Cf. A006496, A045873 (partial sums).

Sequence in context: A119324 A322364 A250034 * A112286 A126261 A050097

Adjacent sequences:  A006492 A006493 A006494 * A006496 A006497 A006498

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Signs from Christian G. Bower, Nov 15 1998

Corrected by Giovanni Resta, Mar 28 2006

STATUS

approved

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Last modified November 14 07:13 EST 2019. Contains 329111 sequences. (Running on oeis4.)