A262710 Powers of -4.

1, -4, 16, -64, 256, -1024, 4096, -16384, 65536, -262144, 1048576, -4194304, 16777216, -67108864, 268435456, -1073741824, 4294967296, -17179869184, 68719476736, -274877906944, 1099511627776, -4398046511104, 17592186044416, -70368744177664, 281474976710656

Offset: 0

Author

L. Van Warren, Sep 28 2015

Keywords

• sign
• easy

Comments

Purely real values from the sequence generated by (1 + i)^k where i = sqrt(-1) and k is a real nonnegative integer.

This sequence gives the values of (1 + i)^k when k is a multiple of 4. When k = 2 mod 4, (1 + i)^k is purely imaginary, and when k is odd, (1 + i)^k has both a real and an imaginary part, and abs(Re((1 + i)^k)) = abs(Im((1 + i)^k)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Caroline Nunn, A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021).
• Index entries for linear recurrences with constant coefficients, signature (-4).

Crossrefs

Cf. A000302, A016825, A005408, A122803.

Programs

Magma

[(-1)^n*4^n: n in [0..30]]; // Vincenzo Librandi, Oct 06 2015

Mathematica

(-4)^Range[0, 15] (* Alonso del Arte, Mar 16 2016 *)

PARI

vector(100, n, n--; (-4)^n) \\ Altug Alkan, Oct 05 2015

PARI

Vec(1/(1+4*x) + O(x^30)) \\ Michel Marcus, Oct 06 2015

PARI

lista(nn) = for (n=0, nn, z = (1+I)^n; if (imag(z)==0, print1(real(z), ", "))); \\ Michel Marcus, Nov 01 2015

Formula

a(n) = (-4)^n.

G.f.: 1/(1 + 4 * x).

E.g.f.: exp(-4*x). - Alejandro J. Becerra Jr., Jan 28 2021

a(n) = Sum_{k=0..2*n} (-1)^k*binomial(4*n, 2*k) (see Nunn, p. 9). - Stefano Spezia, Dec 28 2021