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A321441 Number of "hexagonal partitions" of n.

%I A321441 #18 Jul 23 2025 16:00:09
%S A321441 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,
%T A321441 18,18,20,24,14,21,24,24,22,28,15,26,29,24,22,32,24,31,30,24,26,37,28,
%U A321441 34,29,29,31,46,25,35,36,28,38,45,30,38,42,40,35,46,26
%N A321441 Number of "hexagonal partitions" of n.
%C A321441 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.
%C A321441 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.
%C A321441 The partitions counted here are a superset of those counted at A321440.
%H A321441 Chai Wah Wu, <a href="/A321441/b321441.txt">Table of n, a(n) for n = 0..10000</a>
%e A321441 Here are the derivations of the terms up through n = 10. Partitions are abbreviated as strings of digits.
%e A321441 n = 0: (empty partition)
%e A321441 n = 1: 1
%e A321441 n = 2: 11, 2
%e A321441 n = 3: 111, 12, 3
%e A321441 n = 4: 1111, 112, 22, 4
%e A321441 n = 5: 11111, 122, 23, 5
%e A321441 n = 6: 111111, 1122, 123, 222, 33, 6
%e A321441 n = 7: 1111111, 1222, 223, 34, 7
%e A321441 n = 8: 11111111, 11222, 1223, 2222, 233, 44, 8
%e A321441 n = 9: 111111111, 11223, 12222, 1233, 234, 333, 45, 9
%e A321441 n = 10: 1111111111, 112222, 1234, 22222, 2233, 334, 55, (10)
%o A321441 (Python)
%o A321441 from __future__ import division
%o A321441 def A321441(n):
%o A321441     if n == 0:
%o A321441         return 1
%o A321441     c = 0
%o A321441     for i in range(n):
%o A321441         mi = n + i*(i+1)//2
%o A321441         for j in range(i,n):
%o A321441             mj = mi + j*(j+1)//2
%o A321441             for k in range(j+1,n+1):
%o A321441                 r = mj - k*k
%o A321441                 if r < 0:
%o A321441                     break
%o A321441                 if not r % k:
%o A321441                     c += 1
%o A321441     return c # _Chai Wah Wu_, Nov 11 2018
%Y A321441 Counted partitions are a generalization of those counted at A321440. This sequence has an application to A116513.
%K A321441 nonn
%O A321441 0,3
%A A321441 _Allan C. Wechsler_, Nov 09 2018
%E A321441 More terms from _Chai Wah Wu_, Nov 11 2018