A267313 - OEIS

%I #40 Sep 18 2024 02:48:05

%S 0,-1,1,4,0,5,11,4,12,21,11,22,34,21,35,50,34,51,69,50,70,91,69,92,

%T 116,91,117,144,116,145,175,144,176,209,175,210,246,209,247,286,246,

%U 287,329,286,330,375,329,376,424,375,425,476,424,477,531,476,532,589,531,590,650,589,651,714,650,715

%N Expansion of x*(-1 + 2*x + 3*x^2 - 2*x^3 + x^4)/((1 - x)^3*(1 + x + x^2)^2).

%C First differences are -1, 2, 3, -4, 5, 6, -7, 8, 9, -10, 11, 12, ... - _N. J. A. Sloane_, May 20 2019

%H N. J. A. Sloane, <a href="/A267313/b267313.txt">Table of n, a(n) for n = 0..20000</a>

%H Ilya Gutkovskiy, <a href="/A267313/a267313.pdf">Graphs of first N terms for various values of N</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).

%F G.f.: x*(-1 + 2*x + 3*x^2 - 2*x^3 + x^4)/((1 - x)^3*(1 + x + x^2)^2).

%F a(n) = Sum_{k = 0..n} (-1)^(k mod 3)*k.

%F a(n) = Sum_{k = 0..n} -(-1)^A010882(k)*k.

%F From _G. C. Greubel_, Feb 03 2016: (Start)

%F a(n+7) = a(n+6) + 2*a(n+4) - 2*a(n+3) - a(n+1) + a(n).

%F E.g.f.: (1/18)*exp(-x/2)*((3*x^2+6*x-4)*exp(3*x/2) + 4*(1-3*x)*cos(sqrt(3)*x/2) - 4*sqrt(3)*(1+x)*sin(sqrt(3)*x/2)). (End)

%e a(0) = 0;

%e a(1) = 0 - 1 = -1;

%e a(2) = 0 - 1 + 2 = 1;

%e a(3) = 0 - 1 + 2 + 3 = 4;

%e a(4) = 0 - 1 + 2 + 3 - 4 = 0;

%e a(5) = 0 - 1 + 2 + 3 - 4 + 5 = 5;

%e a(6) = 0 - 1 + 2 + 3 - 4 + 5 + 6 = 11;

%e a(7) = 0 - 1 + 2 + 3 - 4 + 5 + 6 - 7 = 4;

%e a(8) = 0 - 1 + 2 + 3 - 4 + 5 + 6 - 7 + 8 = 12;

%e a(9) = 0 - 1 + 2 + 3 - 4 + 5 + 6 - 7 + 8 + 9 = 21, etc.

%t Table[Sum[(-1)^Mod[k, 3] k, {k, 0, n}], {n, 0, 65}]

%t LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {0, -1, 1, 4, 0, 5, 11}, 65]

%t CoefficientList[Series[x (1 -2 x -3 x^2 +2 x^3 -x^4)/(x^7 -x^6 -2 x^4 + 2 x^3 +x -1), {x,0,70}], x] (* _Vincenzo Librandi_, Jan 13 2016 *)

%o (PARI) Vec(x*(1-2*x-3*x^2+2*x^3-x^4)/(x^7-x^6-2*x^4+2*x^3+x-1) + O(x^100)) \\ _Altug Alkan_, Jan 25 2016

%o (Magma)

%o A267313:= func< n | (&+[(-1)^(k mod 3)*k : k in [0..n]]) >;

%o [A267313(n): n in [0..70]]; // _G. C. Greubel_, Sep 18 2024

%o (SageMath)

%o def A267313(n): return sum((-1)^(k%3)*k for k in range(n+1))

%o [A267313(n) for n in range(71)] # _G. C. Greubel_, Sep 18 2024

%Y Cf. A010882, A130472.

%K sign,easy

%O 0,4

%A _Ilya Gutkovskiy_, Jan 13 2016