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%I #51 Sep 08 2022 08:45:30
%S 0,-1,1,-2,2,-3,3,-4,4,-5,5,-6,6,-7,7,-8,8,-9,9,-10,10,-11,11,-12,12,
%T -13,13,-14,14,-15,15,-16,16,-17,17,-18,18,-19,19,-20,20,-21,21,-22,
%U 22,-23,23,-24,24,-25,25,-26,26,-27,27,-28,28,-29,29,-30,30,-31,31,-32,32
%N A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).
%C Pisano period lengths: 1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, ... - _R. J. Mathar_, Aug 10 2012
%C Partial sums of A038608. - _Stanislav Sykora_, Nov 27 2013
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,1).
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the integers</a>
%F a(n) = -A001057(n).
%F a(2n) = n, a(2n+1) = -(n+1).
%F a(n) = Sum_{k=0..n} k*(-1)^k.
%F a(n) = -a(n-1) +a(n-2) +a(n-3).
%F G.f.: x/( (x-1)*(1+x)^2 ). - _R. J. Mathar_, Feb 20 2011
%F a(n) = floor( (n/2)*(-1)^n ). - _Wesley Ivan Hurt_, Jun 14 2013
%F a(n) = ceiling( n/2 )*(-1)^n. - _Wesley Ivan Hurt_, Oct 22 2013
%F a(n) = ((-1)^n*(2*n+1) - 1)/4. - _Adriano Caroli_, Mar 28 2015
%F E.g.f.: (1/4)*(-exp(x) + (1-2*x)*exp(-x) ). - _G. C. Greubel_, Mar 31 2021
%p A130472:=n->ceil(n/2)*(-1)^n; seq(A130472(k), k=0..50); # _Wesley Ivan Hurt_, Oct 22 2013
%t CoefficientList[Series[(x^2-x)/(1-x^2)^2, {x, 0, 64}], x] (* _Geoffrey Critzer_, Sep 29 2013 *)
%t LinearRecurrence[{-1,1,1},{0,-1,1},70] (* _Harvey P. Dale_, Mar 02 2018 *)
%o (Magma) [((-1)^n*(2*n+1)-1)/4: n in [0..80]]; // _Vincenzo Librandi_, Mar 29 2015
%o (PARI) a(n)=(-1)^n*n\2 \\ _Charles R Greathouse IV_, Sep 02 2015
%o (Sage) [((-1)^n*(2*n+1) - 1)/4 for n in (0..70)] # _G. C. Greubel_, Mar 31 2021
%Y Cf. A059841 (p=0,q=-1), this sequence (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
%Y Cf. A001057, A038608.
%K sign,easy
%O 0,4
%A _Clark Kimberling_, May 28 2007
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