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A307684 Total volume of all rectangular prisms with dimensions r X s X t where r, s and t are the smallest, middle and largest parts in each partition of n into 3 parts. 0
0, 0, 1, 2, 7, 18, 34, 62, 121, 182, 292, 460, 651, 924, 1339, 1764, 2376, 3196, 4074, 5214, 6740, 8294, 10318, 12850, 15496, 18746, 22811, 26936, 32011, 38096, 44376, 51816, 60737, 69768, 80478, 92954, 105735, 120498, 137697, 155078, 175168, 198086, 221452 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..43.

Index entries for sequences related to partitions

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)

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

Cf. A001399.

Sequence in context: A195605 A184096 A136583 * A141631 A172188 A077131

Adjacent sequences:  A307681 A307682 A307683 * A307685 A307686 A307687

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Apr 21 2019

STATUS

approved

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Last modified September 18 19:19 EDT 2020. Contains 337172 sequences. (Running on oeis4.)