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A319007 Sum of the next n nonnegative integers repeated (A004526).

%I A319007 #36 Sep 08 2022 08:46:23
%S A319007 0,1,5,14,29,51,82,124,178,245,327,426,543,679,836,1016,1220,1449,
%T A319007 1705,1990,2305,2651,3030,3444,3894,4381,4907,5474,6083,6735,7432,
%U A319007 8176,8968,9809,10701,11646,12645,13699,14810,15980,17210,18501,19855,21274,22759,24311,25932
%N A319007 Sum of the next n nonnegative integers repeated (A004526).
%C A319007 After 29, all terms are composite.
%H A319007 Colin Barker, <a href="/A319007/b319007.txt">Table of n, a(n) for n = 1..1000</a>
%H A319007 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).
%F A319007 G.f.: x^2*(1 + x + x^2)/((1 + x^2)*(1 - x)^4).
%F A319007 a(n) = -a(-n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 7*a(n-4) + 4*a(n-5) - a(n-6).
%F A319007 a(n) = (2*n*(n^2 - 2) + (1 - (-1)^n)*(-1)^((n-1)/2))/8.
%F A319007 a(n) = A319006(n) - n.
%F A319007 a(n) = (n^3 - 2*n + Chi(n))/4 where Chi(n) = A101455(n). - _Peter Luschny_, Sep 09 2018
%e A319007 Next n nonnegative integers repeated:    Sums:
%e A319007 0,  ......................................   0
%e A319007 0, 1,  ...................................   1
%e A319007 1, 2, 2,  ................................   5
%e A319007 3, 3, 4, 4,  .............................  14
%e A319007 5, 5, 6, 6, 7,  ..........................  29
%e A319007 7, 8, 8, 9, 9, 10,  ......................  51, etc.
%p A319007 a := n -> (n^3 - 2*n + (-(n mod 2))^binomial(n,2))/4;
%p A319007 seq(a(n), n=1..47); # _Peter Luschny_, Sep 09 2018
%t A319007 Table[(2 n (n^2 - 2) + (1 - (-1)^n) (-1)^((n-1)/2))/8, {n, 1, 50}]
%o A319007 (Magma) [Integers()! (n*(n^2-2)+(-(n mod 2))^(n*(n-1)/2))/4: n in [1..50]];
%o A319007 (PARI) concat(0, Vec(x^2*(1 + x + x^2)/((1 + x^2)*(1 - x)^4) + O(x^50))) \\ _Colin Barker_, Sep 10 2018
%Y A319007 Cf. A004526, A101455, A319006.
%Y A319007 Sum of the next n nonnegative integers: A027480.
%K A319007 nonn,easy
%O A319007 1,3
%A A319007 _Bruno Berselli_, Sep 07 2018