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A262710 Powers of -4.

%I A262710 #64 Sep 08 2022 08:46:14
%S A262710 1,-4,16,-64,256,-1024,4096,-16384,65536,-262144,1048576,-4194304,
%T A262710 16777216,-67108864,268435456,-1073741824,4294967296,-17179869184,
%U A262710 68719476736,-274877906944,1099511627776,-4398046511104,17592186044416,-70368744177664,281474976710656
%N A262710 Powers of -4.
%C A262710 Purely real values from the sequence generated by (1 + i)^k where i = sqrt(-1) and k is a real nonnegative integer.
%C A262710 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)).
%H A262710 G. C. Greubel, <a href="/A262710/b262710.txt">Table of n, a(n) for n = 0..1000</a>
%H A262710 Caroline Nunn, <a href="https://scholar.rose-hulman.edu/rhumj/vol22/iss2/3">A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory</a>, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021).
%H A262710 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-4).
%F A262710 a(n) = (-4)^n.
%F A262710 G.f.: 1/(1 + 4 * x).
%F A262710 E.g.f.: exp(-4*x). - _Alejandro J. Becerra Jr._, Jan 28 2021
%F A262710 a(n) = Sum_{k=0..2*n} (-1)^k*binomial(4*n, 2*k) (see Nunn, p. 9). - _Stefano Spezia_, Dec 28 2021
%t A262710 (-4)^Range[0, 15] (* _Alonso del Arte_, Mar 16 2016 *)
%o A262710 (PARI) vector(100, n, n--; (-4)^n) \\ _Altug Alkan_, Oct 05 2015
%o A262710 (PARI) Vec(1/(1+4*x) + O(x^30)) \\ _Michel Marcus_, Oct 06 2015
%o A262710 (PARI) lista(nn) = for (n=0, nn, z = (1+I)^n; if (imag(z)==0, print1(real(z), ", "))); \\ _Michel Marcus_, Nov 01 2015
%o A262710 (Magma) [(-1)^n*4^n: n in [0..30]]; // _Vincenzo Librandi_, Oct 06 2015
%Y A262710 Cf. A000302, A016825, A005408, A122803.
%K A262710 sign,easy
%O A262710 0,2
%A A262710 _L. Van Warren_, Sep 28 2015