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
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Sep 07 2018
STATUS
approved