A321443 Number of "bilaterally symmetric hexagonal partitions" of n.

1, 1, 2, 2, 4, 2, 5, 3, 5, 4, 7, 2, 8, 4, 7, 5, 9, 2, 11, 5, 8, 5, 10, 4, 13, 6, 8, 5, 13, 4, 16, 4, 8, 8, 14, 5, 16, 5, 11, 7, 16, 2, 17, 9, 12, 8, 13, 4, 20, 8, 14, 7, 15, 5, 22, 7, 11, 8, 20, 4, 23, 8, 10, 11, 20, 7, 20, 4, 17, 9, 24, 5, 22, 7, 13, 13, 16

Offset: 0

Author

Allan C. Wechsler, Nov 09 2018

Keywords

• nonn

Comments

A bilaterally symmetric hexagonal partition is one whose parts are consecutive integers, of which all have multiplicity 2 except the largest part, which may have any multiplicity (including 1).

This is a restriction of the concept of hexagonal partition presented in A321441. The nomenclature is suggested by presenting such partitions as hexagonal patches of the triangular lattice A2.

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, 3
n = 4: 1111, 112, 22, 4
n = 5: 11111, 5
n = 6: 111111, 1122, 222, 33, 6
n = 7: 1111111, 223, 7
n = 8: 11111111, 11222, 2222, 44, 8
n = 9: 111111111, 11223, 333, 9
n = 10: 1111111111, 112222, 22222, 2233, 334, 55, (10)
		

Links

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

Crossrefs

A321441 counts hexagonal partitions in general. A321440 counts a different special kind of hexagonal partition. A116513 counts hexagonal "diagrams", of which these partitions are a sort of projection.

Programs

Python

def A321443(n):
    if n == 0:
        return 1
    c = 0
    for i in range(n):
        mi = i*(i+1) + n
        for j in range(i+1,n+1):
            k = mi - j*j
            if k < 0:
                break
            if not k % j:
                c += 1
    return c # Chai Wah Wu, Nov 10 2018

Extensions

More terms from Chai Wah Wu, Nov 10 2018