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A365312
Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.
18
0, 0, 0, 1, 1, 3, 2, 6, 4, 8, 7, 16, 6, 24, 17, 24, 20, 46, 22, 62, 31, 63, 57, 106, 35, 122, 90, 137, 88, 212, 74, 262, 134, 267, 206, 345, 121, 476, 294, 484, 232, 698, 242, 837, 389, 763, 571, 1185, 318, 1327, 634, 1392, 727, 1927, 640, 2056, 827, 2233, 1328
OFFSET
0,6
EXAMPLE
The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
The a(0) = 0 through a(11) = 16 strict partitions:
. . . (2) (3) (2) (4) (2) (3) (2) (3) (2)
(3) (5) (3) (5) (4) (4) (3)
(4) (4) (6) (5) (6) (4)
(5) (7) (6) (7) (5)
(6) (7) (8) (6)
(4,2) (8) (9) (7)
(4,2) (6,3) (8)
(6,2) (9)
(10)
(4,2)
(5,4)
(6,2)
(6,3)
(6,4)
(7,3)
(8,2)
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions, n], UnsameQ@@#&], combs[n, #]=={}&]], {n, 0, 10}]
PROG
(Python)
from math import isqrt
from sympy.utilities.iterables import partitions
def A365312(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1, n+1) for b in partitions(m, m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
CROSSREFS
The complement for positive coefficients is counted by A088314.
For positive coefficients we have A088528.
The complement is counted by A365311.
For non-strict partitions we have A365378, complement A365379.
The version for subsets is A365380, complement A365073.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Sequence in context: A321508 A245261 A054089 * A365080 A257903 A257877
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 05 2023
EXTENSIONS
a(26)-a(58) from Chai Wah Wu, Sep 13 2023
STATUS
approved