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

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

Author

N. J. A. Sloane

Keywords

• sign
• easy

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).

Crossrefs

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

Programs

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

Mathematica

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

PARI

a(n)=([1, -2; 2, 1]^n)[1,2] \\ Charles R Greathouse IV, Dec 22 2011

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)

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

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