OFFSET
0,2
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).
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
MATHEMATICA
(-4)^Range[0, 15] (* Alonso del Arte, Mar 16 2016 *)
PROG
(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
(Magma) [(-1)^n*4^n: n in [0..30]]; // Vincenzo Librandi, Oct 06 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
L. Van Warren, Sep 28 2015
STATUS
approved