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A321443 Number of "bilaterally symmetric hexagonal partitions" of n 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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, 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)

PROG

(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

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.

Sequence in context: A187782 A129296 A300837 * A125296 A294097 A233520

Adjacent sequences:  A321440 A321441 A321442 * A321444 A321445 A321446

KEYWORD

nonn

AUTHOR

Allan C. Wechsler, Nov 09 2018

EXTENSIONS

More terms from Chai Wah Wu, Nov 10 2018

STATUS

approved

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Last modified October 19 01:25 EDT 2019. Contains 328211 sequences. (Running on oeis4.)