A319006 Sum of the next n positive integers repeated (A008619).

1, 3, 8, 18, 34, 57, 89, 132, 187, 255, 338, 438, 556, 693, 851, 1032, 1237, 1467, 1724, 2010, 2326, 2673, 3053, 3468, 3919, 4407, 4934, 5502, 6112, 6765, 7463, 8208, 9001, 9843, 10736, 11682, 12682, 13737, 14849, 16020, 17251, 18543, 19898, 21318, 22804, 24357, 25979

Offset

1,2

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*(1 - x + 3*x^2 - x^3 + x^4)/((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) = A319007(n) + n.

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

Example

Next n positive integers repeated:       Sums:
1,  ......................................   1
1, 2,  ...................................   3
2, 3, 3,  ................................   8
4, 4, 5,  5,  ............................  18
6, 6, 7,  7,  8,  ........................  34
8, 9, 9, 10, 10, 11,  ....................  57, 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}]
Module[{nn=50, lst}, lst=Flatten[Table[{n, n}, {n, (nn(nn+1))/2}]]; Total/@ TakeList[lst, Range[nn]]] (* Requires Mathematica version 11 or later *) (* or *) LinearRecurrence[{4, -7, 8, -7, 4, -1}, {1, 3, 8, 18, 34, 57}, 50] (* Harvey P. Dale, Jul 10 2021 *)

Prog

(Magma) [Integers()! (n*(n^2+2)+(-(n mod 2))^(n*(n-1)/2))/4: n in [1..50]];
(PARI) Vec(x*(1 - x + 3*x^2 - x^3 + x^4)/((1 + x^2)*(1 - x)^4) + O(x^50)) \\ Colin Barker, Sep 10 2018

Crossrefs

Cf. A008619, A101455, A319007.

Sum of the next n positive integers: A006003 (after 0).

Sequence in context: A184636 A075342 A083726 * A212589 A367188 A081489

Adjacent sequences: A319003 A319004 A319005 * A319007 A319008 A319009

Keyword

nonn,easy

Author

Bruno Berselli, Sep 07 2018

Status

approved