A343401 Number of partitions of n into 3 parts [r,s,t] such that r < s < t where no part divides any larger part and at least 1 part divides n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 4, 0, 7, 0, 9, 7, 6, 0, 17, 5, 8, 8, 18, 0, 28, 0, 20, 13, 13, 16, 44, 0, 13, 18, 46, 0, 50, 0, 36, 37, 18, 0, 77, 14, 45, 24, 47, 0, 71, 34, 78, 31, 24, 0, 134, 0, 24, 65, 74, 38, 96, 0, 68, 37, 108, 0, 174, 0, 31, 80, 76, 51

Offset

1,12

Links

Table of n, a(n) for n=1..77.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} sign(3 - c(n/j) - c(n/i) - c(n/(n-i-j))) * c(i/j) * c((n-i-j)/i) * c((n-i-j)/j), where c(n) = ceiling(n) - floor(n).

Example

a(12) = 2; [2,3,7], [3,4,5] (Not counted [1,1,10], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,2,8], [2,4,6], [2,5,5], [3,3,6], [4,4,4]).

Mathematica

Table[Sum[Sum[Sign[3 - Ceiling[n/j] + Floor[n/j] - Ceiling[n/(n - i - j)] + Floor[n/(n - i - j)] - Ceiling[n/i] + Floor[n/i]] (Ceiling[i/j] - Floor[i/j]) (Ceiling[(n - i - j)/j] - Floor[(n - i - j)/j]) (Ceiling[(n - i - j)/i] - Floor[(n - i - j)/i]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]

Prog

(PARI) c(n) = ceil(n)-floor(n)
a(n) = sum(j=1, floor(n/3), sum(i=j, floor((n-j)/2), sign(3-c(n/j)-c(n/i)-c(n/(n-i-j))) * c(i/j) * c((n-i-j)/i) * c((n-i-j)/j))) \\ Felix Fröhlich, Apr 21 2021

Crossrefs

Cf. A343246.

Sequence in context: A238158 A029906 A094907 * A226570 A158380 A051734

Adjacent sequences: A343398 A343399 A343400 * A343402 A343403 A343404

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 13 2021

Status

approved