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A076118 a(n) = Sum_{k=n/2..n} k * (-1)^(n-k) * C(k,n-k).

%I A076118 #36 May 10 2025 03:13:13
%S A076118 0,1,1,-1,-3,-2,2,5,3,-3,-7,-4,4,9,5,-5,-11,-6,6,13,7,-7,-15,-8,8,17,
%T A076118 9,-9,-19,-10,10,21,11,-11,-23,-12,12,25,13,-13,-27,-14,14,29,15,-15,
%U A076118 -31,-16,16,33,17,-17,-35,-18,18,37,19,-19,-39,-20,20,41,21,-21,-43,-22,22,45,23,-23,-47,-24,24,49,25,-25,-51,-26,26
%N A076118 a(n) = Sum_{k=n/2..n} k * (-1)^(n-k) * C(k,n-k).
%C A076118 Piecewise linear depending on residue modulo 6. Might be described as an inverse Catalan transform of the nonnegative integers.
%C A076118 Number of compositions of n consisting of at most two parts, all congruent to {0,2} mod 3 (offset 1). - _Vladeta Jovovic_, Mar 10 2005
%H A076118 Robert Israel, <a href="/A076118/b076118.txt">Table of n, a(n) for n = 0..10000</a>
%H A076118 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,2,-1).
%F A076118 a(3n) = -a(3n-1) = A038608(n).
%F A076118 a(n) = ( 2n*sin((n+1/2)*Pi/3) + sin(n*Pi/3)/sin(Pi/3) )/3.
%F A076118 a(3n) = n*(-1)^n; a(3n+1) = (2n+1)*(-1)^n; a(3n+2) = (n+1)*(-1)^n.
%F A076118 a(n) = Sum{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*(n-k). - _Paul Barry_, Nov 12 2004
%F A076118 From _Michael Somos_, Jul 14 2006: (Start)
%F A076118 Euler transform of length 6 sequence [ 1, -2, -2, 0, 0, 2].
%F A076118 G.f.: x(1-x)/(1-x+x^2)^2 = x*(1-x^2)^2*(1-x^3)^2/((1-x)*(1-x^6)^2).
%F A076118 a(-1-n)=a(n). (End)
%F A076118 a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n). - _Robert Israel_, Aug 07 2015
%F A076118 a(n) = A099254(n-1)-A099254(n-2). - _R. J. Mathar_, Apr 01 2018
%F A076118 Sum_{n>=1} 1/a(n) = Pi/4 (A003881). - _Amiram Eldar_, May 10 2025
%e A076118 a(10) = -5*1 + 6*15 - 7*35 + 8*28 - 9*9 + 10*1 = -5 + 90 -245 + 224 - 81 + 10 = -7.
%p A076118 A076118:=n->add(k*(-1)^(n-k)*binomial(k,n-k), k=floor(n/2)..n); seq(A076118(n), n=0..50); # _Wesley Ivan Hurt_, May 08 2014
%p A076118 f:= gfun:-rectoproc({a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n), a(0)=0,a(1)=1,a(2)=1,a(3)=-1}, a(n), remember):
%p A076118 map(f, [$0..100]); # _Robert Israel_, Aug 07 2015
%t A076118 Table[Sum[k*(-1)^(n - k)*Binomial[k, n - k], {k, Floor[n/2], n}], {n,
%t A076118 0, 50}] (* _Wesley Ivan Hurt_, May 08 2014 *)
%o A076118 (PARI) {a(n)=local(k=n%3); n=n\3; (-1)^n*((k>0)+n+(k==1)*n)} /* _Michael Somos_, Jul 14 2006 */
%o A076118 (PARI) {a(n)=if(n<0, n=-1-n); polcoeff(x*(1-x)/(1-x+x^2)^2+x*O(x^n),n)} /* _Michael Somos_, Jul 14 2006 */
%Y A076118 Cf. A003881, A038608, A078028, A099254 (partial sums).
%Y A076118 See A151842 for a version without signs.
%K A076118 sign,easy
%O A076118 0,5
%A A076118 _Henry Bottomley_, Oct 31 2002