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A365659
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Number of strict integer partitions of n that either have (1) length 2, or (2) greatest part n/2.
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7
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0, 0, 0, 1, 1, 2, 3, 3, 4, 4, 6, 5, 8, 6, 10, 7, 12, 8, 15, 9, 18, 10, 21, 11, 25, 12, 29, 13, 34, 14, 40, 15, 46, 16, 53, 17, 62, 18, 71, 19, 82, 20, 95, 21, 109, 22, 125, 23, 144, 24, 165, 25, 189, 26, 217, 27, 248, 28, 283, 29, 324
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OFFSET
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0,6
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COMMENTS
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Also the number of strict integer partitions of n containing two possibly equal elements summing to n.
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LINKS
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FORMULA
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a(n) = (n-1)/2 if n is odd. a(n) = n/2 + A000009(n/2) - 2 if n is even and n > 0. - Chai Wah Wu, Sep 18 2023
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EXAMPLE
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The a(3) = 1 through a(11) = 5 partitions:
(2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4) (6,4) (6,5)
(4,1) (5,1) (5,2) (6,2) (6,3) (7,3) (7,4)
(3,2,1) (6,1) (7,1) (7,2) (8,2) (8,3)
(4,3,1) (8,1) (9,1) (9,2)
(5,3,2) (10,1)
(5,4,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(Length[#]==2||Max@@#==n/2)&]], {n, 0, 30}]
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PROG
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(Python)
from sympy.utilities.iterables import partitions
def A365659(n): return n>>1 if n&1 or n==0 else (m:=n>>1)+sum(1 for p in partitions(m) if max(p.values(), default=1)==1)-2 # Chai Wah Wu, Sep 18 2023
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CROSSREFS
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Without repeated parts we have A140106.
For subsets instead of strict partitions we have A365544.
A000009 counts subsets summing to n.
A365543 counts partitions of n with a submultiset summing to k.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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