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.

LINKS

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

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

KEYWORD

nonn

AUTHOR

Allan C. Wechsler, Nov 09 2018

EXTENSIONS

More terms from Chai Wah Wu, Nov 11 2018

STATUS

approved