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A225883 a(n) = (-1)^n * (1 - 2^n).

%I A225883 #19 May 16 2025 09:18:18
%S A225883 0,1,-3,7,-15,31,-63,127,-255,511,-1023,2047,-4095,8191,-16383,32767,
%T A225883 -65535,131071,-262143,524287,-1048575,2097151,-4194303,8388607,
%U A225883 -16777215,33554431,-67108863,134217727,-268435455,536870911,-1073741823,2147483647,-4294967295
%N A225883 a(n) = (-1)^n * (1 - 2^n).
%C A225883 A001047(n) = p(0) where p(x) is the unique degree-(n-1) polynomial such that p(k) = a(k) for k = 1, 2, ..., n.
%C A225883 a(n) = p(n) where p(x) is the unique degree-(n-1) polynomial such that p(k) = (-1)^k for k = 0, 1, ..., n-1.
%H A225883 G. C. Greubel, <a href="/A225883/b225883.txt">Table of n, a(n) for n = 0..2500</a>
%H A225883 Bérénice Delcroix-Oger and Clément Dupont, <a href="https://arxiv.org/abs/2505.06094">Lie-operads and operadic modules from poset cohomology</a>, arXiv:2505.06094 [math.CO], 2025. See p. 28.
%H A225883 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2).
%F A225883 G.f.: x / ((1 + x) * (1 + 2*x)) = 1 / (1+x) - 1 / (1 + 2*x).
%F A225883 E.g.f.: exp(-x) - exp(-2*x). a(n) = -2 * a(n-1) - (-1)^n if n>0.
%F A225883 a(n) = -(-1)^n * A000225(n). a(n) = -3 * a(n-1) - 2 * a(n-2) if n>1.
%F A225883 REVERT transform is A001003 omitting a(0)=0.
%F A225883 INVERT transform is A108520.
%F A225883 2^n = a(n+1)^2 - a(n) * a(n+2).
%F A225883 |a(n)| = A168604(n+2)= A000225(n).
%e A225883 G.f. = x - 3*x^2 + 7*x^3 - 15*x^4 + 31*x^5 - 63*x^6 + 127*x^7 - 255*x^8 + 511*x^9 + ...
%t A225883 a[ n_] := If[ n<0, 0, (-1)^n (1 - 2^n)];
%t A225883 LinearRecurrence[{-3,-2}, {0,1}, 50] (* _G. C. Greubel_, Aug 09 2018 *)
%o A225883 (PARI) {a(n) = if( n<0, 0, (-1)^n * (1 - 2^n))};
%o A225883 (PARI) {a(n) = if( n<0, 0, polcoeff( x / ((1 + x) * (1 + 2*x)) + x * O(x^n), n))};
%o A225883 (Magma) [(-1)^n*(1 - 2^n): n in [0..50]]; // _G. C. Greubel_, Aug 09 2018
%Y A225883 Cf. A000225, A001003, A001047, A108520.
%K A225883 sign,easy
%O A225883 0,3
%A A225883 _Michael Somos_, May 19 2013