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

%I A321443 #11 Jul 23 2025 16:00:16
%S A321443 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 A321443 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 A321443 23,8,10,11,20,7,20,4,17,9,24,5,22,7,13,13,16
%N A321443 Number of "bilaterally symmetric hexagonal partitions" of n.
%C A321443 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 A321443 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 A321443 Chai Wah Wu, <a href="/A321443/b321443.txt">Table of n, a(n) for n = 0..10000</a>
%e A321443 Here are the derivations of the terms up through n = 10. Partitions are abbreviated as strings of digits.
%e A321443 n = 0: (empty partition)
%e A321443 n = 1: 1
%e A321443 n = 2: 11, 2
%e A321443 n = 3: 111, 3
%e A321443 n = 4: 1111, 112, 22, 4
%e A321443 n = 5: 11111, 5
%e A321443 n = 6: 111111, 1122, 222, 33, 6
%e A321443 n = 7: 1111111, 223, 7
%e A321443 n = 8: 11111111, 11222, 2222, 44, 8
%e A321443 n = 9: 111111111, 11223, 333, 9
%e A321443 n = 10: 1111111111, 112222, 22222, 2233, 334, 55, (10)
%o A321443 (Python)
%o A321443 def A321443(n):
%o A321443     if n == 0:
%o A321443         return 1
%o A321443     c = 0
%o A321443     for i in range(n):
%o A321443         mi = i*(i+1) + n
%o A321443         for j in range(i+1,n+1):
%o A321443             k = mi - j*j
%o A321443             if k < 0:
%o A321443                 break
%o A321443             if not k % j:
%o A321443                 c += 1
%o A321443     return c # _Chai Wah Wu_, Nov 10 2018
%Y A321443 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 A321443 nonn
%O A321443 0,3
%A A321443 _Allan C. Wechsler_, Nov 09 2018
%E A321443 More terms from _Chai Wah Wu_, Nov 10 2018