A321443 - OEIS

%I #9 Nov 10 2018 10:14:09

%S 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,

%T 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,

%U 23,8,10,11,20,7,20,4,17,9,24,5,22,7,13,13,16

%N Number of "bilaterally symmetric hexagonal partitions" of n

%C 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).

%C 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.

%H Chai Wah Wu, <a href="/A321443/b321443.txt">Table of n, a(n) for n = 0..10000</a>

%e Here are the derivations of the terms up through n = 10. Partitions are abbreviated as strings of digits.

%e n = 0: (empty partition)

%e n = 1: 1

%e n = 2: 11, 2

%e n = 3: 111, 3

%e n = 4: 1111, 112, 22, 4

%e n = 5: 11111, 5

%e n = 6: 111111, 1122, 222, 33, 6

%e n = 7: 1111111, 223, 7

%e n = 8: 11111111, 11222, 2222, 44, 8

%e n = 9: 111111111, 11223, 333, 9

%e n = 10: 1111111111, 112222, 22222, 2233, 334, 55, (10)

%o (Python)

%o def A321443(n):

%o     if n == 0:

%o         return 1

%o     c = 0

%o     for i in range(n):

%o         mi = i*(i+1) + n

%o         for j in range(i+1,n+1):

%o             k = mi - j*j

%o             if k < 0:

%o                 break

%o             if not k % j:

%o                 c += 1

%o     return c # _Chai Wah Wu_, Nov 10 2018

%Y 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.

%K nonn

%O 0,3

%A _Allan C. Wechsler_, Nov 09 2018

%E More terms from _Chai Wah Wu_, Nov 10 2018