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
Examples
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.
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).
Programs
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Magma
[Integers()! (n*(n^2+2)+(-(n mod 2))^(n*(n-1)/2))/4: n in [1..50]];
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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
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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 *) -
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
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