A006496 Imaginary part of (1+2i)^n. (Formerly M0933)

0, 2, 4, -2, -24, -38, 44, 278, 336, -718, -3116, -2642, 10296, 33802, 16124, -136762, -354144, -24478, 1721764, 3565918, -1476984, -20783558, -34182196, 35553398, 242017776, 306268562, -597551756, -2726446322, -2465133864, 8701963882, 29729597084, 15949374758

Offset

0,2

Comments

The absolute values of these numbers are the even numbers x such that x^2 + y^2 = 5^n with x and y coprime. See A098122. - T. D. Noe, Apr 14 2011

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

George Berzsenyi, Gaussian Fibonacci numbers, Fib. Quart., Vol. 15, No. 3 (1977), pp. 233-236.

Index entries for Gaussian integers and primes.

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

Formula

a(n) = 2*a(n-1) - 5*a(n-2); a(0)=0, a(1)=2. - T. D. Noe, Nov 09 2006

a(n) = - [M^n]_1,2, where M = [1, -2; 2, 1]. - Simone Severini, Apr 25 2007

A000351(n) = A006495(n)^2 + a(n)^2. - Fabrice Baubet, May 28 2007

From R. J. Mathar, Apr 06 2008: (Start)

O.g.f.: 2*x/(1 - 2*x + 5*x^2).

a(n) = 2*A045873(n). (End)

a(n) = (1/2)*i*(1-2*i)^n - (1/2)*i*(1+2*i)^n, with n>=0 and i=sqrt(-1). - Paolo P. Lava, Oct 03 2008

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

a(n)/A006495(n) = -tan(2*n*arctan(phi)), where phi is the golden ratio (A001622). - Amiram Eldar, Jan 13 2022

Mathematica

LinearRecurrence[{2, -5}, {0, 2}, 30] (* Vincenzo Librandi, Dec 21 2011 *)

Prog

(Magma) I:=[0, 2]; [n le 2 select I[n] else 2*Self(n-1)-5*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 21 2011
(PARI) a(n)=([1, -2; 2, 1]^n)[1, 2] \\ Charles R Greathouse IV, Dec 22 2011

Crossrefs

Cf. A000351, A001622, A006495, A098122, A045873.

Sequence in context: A059890 A295640 A210457 * A352642 A361077 A263931

Adjacent sequences: A006493 A006494 A006495 * A006497 A006498 A006499

Keyword

sign,easy

Author

N. J. A. Sloane

Extensions

Signs from Christian G. Bower, Nov 15 1998

Corrected by T. D. Noe, Nov 09 2006

More terms from R. J. Mathar, Apr 06 2008

Status

approved