[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A262710
Powers of -4.
7
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
0,2
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
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).
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