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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 (list; graph; refs; listen; history; text; internal format)
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

Counted partitions are a generalization of those counted at A312440. This sequence has an application to A116513.

Sequence in context: A158973 A071323 A071324 * A063655 A111234 A117248

Adjacent sequences:  A321438 A321439 A321440 * A321442 A321443 A321444

KEYWORD

nonn

AUTHOR

Allan C. Wechsler, Nov 09 2018

EXTENSIONS

More terms from Chai Wah Wu, Nov 11 2018

STATUS

approved

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Last modified June 24 03:27 EDT 2021. Contains 345415 sequences. (Running on oeis4.)