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

0, 1, 5, 14, 29, 51, 82, 124, 178, 245, 327, 426, 543, 679, 836, 1016, 1220, 1449, 1705, 1990, 2305, 2651, 3030, 3444, 3894, 4381, 4907, 5474, 6083, 6735, 7432, 8176, 8968, 9809, 10701, 11646, 12645, 13699, 14810, 15980, 17210, 18501, 19855, 21274, 22759, 24311, 25932

Offset

1,3

Comments

After 29, all terms are composite.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-7,4,-1).

Formula

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

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).

a(n) = (2*n*(n^2 - 2) + (1 - (-1)^n)*(-1)^((n-1)/2))/8.

a(n) = A319006(n) - n.

a(n) = (n^3 - 2*n + Chi(n))/4 where Chi(n) = A101455(n). - Peter Luschny, Sep 09 2018

Example

Next n nonnegative integers repeated:    Sums:
0,  ......................................   0
0, 1,  ...................................   1
1, 2, 2,  ................................   5
3, 3, 4, 4,  .............................  14
5, 5, 6, 6, 7,  ..........................  29
7, 8, 8, 9, 9, 10,  ......................  51, etc.

Maple

a := n -> (n^3 - 2*n + (-(n mod 2))^binomial(n, 2))/4;
seq(a(n), n=1..47); # Peter Luschny, Sep 09 2018

Mathematica

Table[(2 n (n^2 - 2) + (1 - (-1)^n) (-1)^((n-1)/2))/8, {n, 1, 50}]

Prog

(Magma) [Integers()! (n*(n^2-2)+(-(n mod 2))^(n*(n-1)/2))/4: n in [1..50]];
(PARI) concat(0, Vec(x^2*(1 + x + x^2)/((1 + x^2)*(1 - x)^4) + O(x^50))) \\ Colin Barker, Sep 10 2018

Crossrefs

Cf. A004526, A101455, A319006.

Sum of the next n nonnegative integers: A027480.

Sequence in context: A005918 A321178 A256666 * A211651 A299291 A019262

Adjacent sequences: A319004 A319005 A319006 * A319008 A319009 A319010

Keyword

nonn,easy

Author

Bruno Berselli, Sep 07 2018

Status

approved