OFFSET
1,4
LINKS
Sela Fried, On the ordinary generating function of A307684, 2024.
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 10.
FORMULA
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} i * k * (n-i-k).
Conjectures from Colin Barker, Apr 22 2019: (Start)
G.f.: x^3*(1 + 3*x + 7*x^2 + 15*x^3 + 23*x^4 + 21*x^5 + 18*x^6 + 14*x^7 + 6*x^8) / ((1 - x)^6*(1 + x)^3*(1 + x + x^2)^4).
a(n) = -a(n-1) + 2*a(n-2) + 6*a(n-3) + 3*a(n-4) - 9*a(n-5) - 14*a(n-6) - 2*a(n-7) + 16*a(n-8) + 16*a(n-9) - 2*a(n-10) - 14*a(n-11) - 9*a(n-12) + 3*a(n-13) + 6*a(n-14) + 2*a(n-15) - a(n-16) - a(n-17) for n>17. (End)
These conjectures are true. See links. - Sela Fried, Aug 06 2024
EXAMPLE
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
-----------------------------------------------------------------------
n | 3 4 5 6 7 8 9 10 ...
-----------------------------------------------------------------------
a(n) | 1 2 7 18 34 62 121 182 ...
-----------------------------------------------------------------------
MATHEMATICA
Table[Sum[Sum[i*k*(n - i - k), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 100}]
PROG
(PARI) a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, i*k*(n-i-k))); \\ Michel Marcus, Apr 22 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 21 2019
STATUS
approved