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A321441
Number of "hexagonal partitions" of n
3
1, 1, 2, 3, 4, 4, 6, 5, 7, 8, 8, 7, 11, 8, 12, 12, 11, 10, 16, 12, 15, 15, 14, 14, 22, 14, 18, 18, 18, 20, 24, 14, 21, 24, 24, 22, 28, 15, 26, 29, 24, 22, 32, 24, 31, 30, 24, 26, 37, 28, 34, 29, 29, 31, 46, 25, 35, 36, 28, 38, 45, 30, 38, 42, 40, 35, 46, 26
OFFSET
0,3
COMMENTS
A "hexagonal partition" is one whose parts are consecutive, whose largest part has arbitrary multiplicity, and the remaining parts have multiplicity 1 or 2, with the single parts smaller than the double parts.
Each of the hexagonal diagrams counted by A116513 corresponds to at most three of these partitions, so this sequence is bounded above by 3*A116513. The relationship between A116513 and this sequence would bear further study.
The partitions counted here are a superset of those counted at A321440.
EXAMPLE
Here are the derivations of the terms up through n = 10. Partitions are abbreviated as strings of digits.
n = 0: (empty partition)
n = 1: 1
n = 2: 11, 2
n = 3: 111, 12, 3
n = 4: 1111, 112, 22, 4
n = 5: 11111, 122, 23, 5
n = 6: 111111, 1122, 123, 222, 33, 6
n = 7: 1111111, 1222, 223, 34, 7
n = 8: 11111111, 11222, 1223, 2222, 233, 44, 8
n = 9: 111111111, 11223, 12222, 1233, 234, 333, 45, 9
n = 10: 1111111111, 112222, 1234, 22222, 2233, 334, 55, (10)
PROG
(Python)
from __future__ import division
def A321441(n):
if n == 0:
return 1
c = 0
for i in range(n):
mi = n + i*(i+1)//2
for j in range(i, n):
mj = mi + j*(j+1)//2
for k in range(j+1, n+1):
r = mj - k*k
if r < 0:
break
if not r % k:
c += 1
return c # Chai Wah Wu, Nov 11 2018
CROSSREFS
Counted partitions are a generalization of those counted at A321440. This sequence has an application to A116513.
Sequence in context: A071323 A071324 A361003 * A063655 A111234 A117248
KEYWORD
nonn
AUTHOR
Allan C. Wechsler, Nov 09 2018
EXTENSIONS
More terms from Chai Wah Wu, Nov 11 2018
STATUS
approved