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URL: https://oeis.org/A343401

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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, 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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Apr 13 2021
STATUS
approved