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%I #62 Dec 18 2023 19:31:19
%S 1,-2,4,-8,16,-32,64,-128,256,-512,1024,-2048,4096,-8192,16384,-32768,
%T 65536,-131072,262144,-524288,1048576,-2097152,4194304,-8388608,
%U 16777216,-33554432,67108864,-134217728,268435456,-536870912,1073741824,-2147483648,4294967296,-8589934592,17179869184
%N Powers of -2: a(n) = (-2)^n.
%C The number -2 can be used as a base of numeration (see the Weisstein link). - _Alonso del Arte_, Mar 30 2014
%C Contribution from _M. F. Hasler_, Oct 21 2014: (Start)
%C This is the inverse binomial transform of A033999 = n->(-1)^n, and the binomial transform of A033999*A000244 = n->(-3)^n, see also A141413.
%C Prefixed with one 0, i.e., (0,1,-2,4,...) = -A033999*A131577, it is the binomial transform of (0, 1, -4, 13, -40, 121,...) = -A033999*A003462, and inverse binomial transform of (0,1,0,1,0,1,...) = A000035.
%C Prefixed with two 0's, i.e., (0,0,1,-2,4,-8,...), it is the binomial transform of (0,0,1,-5,18,-58,179,-543,...) (cf. A000340) and inverse binomial transform of (0,0,1,1,2,2,3,3,...) = A004526. (End)
%C Prefixed with three 0's, this is the inverse binomial difference of (0, 0, 0, 1, 2, 4, 6, 9, 12, 16,...) = concat(0, A002620), which has as successive differences (0, 0, 1, 1, 2, 2,...) = A004526, then (0, 1, 0, 1,...) = A000035, then (1, -1, 1, -1,...) = A033999, and then (-2)^k*A033999 with k=1,2,3,... - _Paul Curtz_, Oct 16 2014, edited by _M. F. Hasler_, Oct 21 2014
%C Stirling-Bernoulli transform of triangular numbers: 1, 3, 6, 10, 15, 21, 28, ... - _Philippe Deléham_, May 25 2015
%H Franklin T. Adams-Watters, <a href="/A122803/b122803.txt">Table of n, (-2)^n for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Negabinary.html">Negabinary</a>.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-2).
%F a(n) = (-2)^n = (-1)^n * 2^n.
%F a(n) = -2*a(n-1), n > 0; a(0) = 1. G.f.: 1/(1+2x). - _Philippe Deléham_, Nov 19 2008
%F Sum_{n >= 0} 1/a(n) = 2/3. - _Jaume Oliver Lafont_, Mar 01 2009
%F E.g.f.: 1/exp(2*x). - _Arkadiusz Wesolowski_, Aug 13 2012
%F a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A030195(n+1). - _R. J. Mathar_, Oct 15 2012
%F G.f.: 1/(1+2x). A122803 = A033999 * A000079. - _M. F. Hasler_, Oct 21 2014
%F a(n) = Sum_{k = 0..n} A163626(n,k)*A000217(k+1). - _Philippe Deléham_, May 25 2015
%p A122803:=n->(-2)^n; seq(A122803(n), n=0..50); # _Wesley Ivan Hurt_, Mar 30 2014
%t Table[(-2)^n, {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2011 *)
%o (Magma) [(-2)^n: n in [0..60]]; // _Vincenzo Librandi_, Oct 22 2014
%o (PARI) a(n)=(-2)^n \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Python)
%o def A122803(n): return -(1<<n) if n&1 else 1<<n # _Chai Wah Wu_, Nov 18 2022
%Y Cf. A000079 (powers of 2), A004526, A005351, A005352, A011782, A034008, A131577, A171449, A248800.
%K easy,sign
%O 0,2
%A _Franklin T. Adams-Watters_, Sep 11 2006
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