A139011 Real part of (2 + i)^n, where i = sqrt(-1).

1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882

Offset: 0

Author

Gary W. Adamson, Apr 05 2008

Keywords

• sign
• easy

Comments

Imaginary part of (2 + i)^n gives A099456.

Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495.

Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008

Examples

			1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ...
a(5) = -38 since (2 + i)^5 = (-38 + 41*i).
a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5).
a(5) = -38 = A006496(5).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2862 (first 201 terms from Vincenzo Librandi)
• Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
• Index entries for linear recurrences with constant coefficients, signature (4,-5).

Crossrefs

Cf. A099456, A006495, A006496, A056594, A146559 (inv bin. transf.).

Programs

Magma

[ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011

Maple

restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..29 ); # Zerinvary Lajos, Apr 06 2009

Mathematica

Re[(2+I)^Range[0,30]] (* or *) LinearRecurrence[{4,-5},{1,2},30] (* Harvey P. Dale, Nov 02 2022 *)

PARI

a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */

PARI

Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

Sage

[lucas_number2(n,4,5)/2 for n in range(0,31)] # Zerinvary Lajos, Jul 08 2008

Formula

Real part of (2 + i)^n, i^2 = -1.

Term (1,1) of matrix [2,-1; 1,2]^n.

(a(n))^2 + (A099456(n))^2 = 5^n.

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

O.g.f.: (1-2x) /(1-4x+5x^2).

a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End)

E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009

a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010

a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008

2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011

a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*binomial(n,2*k). - Gerry Martens, Sep 18 2022

Extensions

Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009

Added a(0)=1 by Michael Somos, Dec 26 2010

Edited by Franklin T. Adams-Watters, Apr 10 2011