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A137501 The even numbers repeated, with alternating signs.

%I A137501 #30 Feb 27 2019 01:19:09
%S A137501 0,0,2,-2,4,-4,6,-6,8,-8,10,-10,12,-12,14,-14,16,-16,18,-18,20,-20,22,
%T A137501 -22,24,-24,26,-26,28,-28,30,-30,32,-32,34,-34,36,-36,38,-38,40,-40,
%U A137501 42,-42,44,-44,46,-46,48,-48,50,-50,52,-52,54,-54,56,-56,58,-58,60,-60,62,-62,64,-64,66,-66,68,-68,70,-70,72,-72,74
%N A137501 The even numbers repeated, with alternating signs.
%C A137501 The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=1 and k shifted one place, thus a(k) = (P(1,1)-(-1)^(k-1) P(1,2(k-1)+1))/2. - _Peter Luschny_, Jul 12 2009
%C A137501 With just one 0 at the beginning, this is a permutation of all the even integers. - _Alonso del Arte_, Jun 24 2012
%H A137501 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,1).
%F A137501 a(n) = ( n - (1/2) + (1/2)*(-1)^n )*(-1)^n.
%F A137501 From _R. J. Mathar_, Feb 14 2010: (Start)
%F A137501 a(n) = -a(n-1) + a(n-2) + a(n-3).
%F A137501 G.f.: 2*x^2/((1-x) * (1+x)^2). (End)
%F A137501 a(n) = A064455(n) - A123684(n). - _Jaroslav Krizek_, Mar 22 2011
%p A137501 den:= n -> (n-1/2+1/2*(-1)^n)*(-1)^n: seq(den(n),n=-10..10);
%p A137501 a := n -> (1+(-1)^n*(2*n-1))/2; # _Peter Luschny_, Jul 12 2009
%t A137501 Flatten[Table[{2n, -2n}, {n, 0, 39}]] (* _Alonso del Arte_, Jun 24 2012 *)
%t A137501 With[{enos=2*Range[0,40]},Riffle[enos,-enos]] (* _Harvey P. Dale_, Oct 12 2014 *)
%Y A137501 Cf. A052928, A064455, A123684, A153641.
%K A137501 easy,sign
%O A137501 0,3
%A A137501 Carlos Alberto da Costa Filho (cacau_dacosta(AT)hotmail.com), Apr 22 2008