A321441 Number of "hexagonal partitions" of n.

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

Author

Allan C. Wechsler, Nov 09 2018

Keywords

• nonn

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.

Examples

			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)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000

Crossrefs

Counted partitions are a generalization of those counted at A321440. This sequence has an application to A116513.

Programs

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

Extensions

More terms from Chai Wah Wu, Nov 11 2018