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 A363226 Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions. 24
 0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 5, 4, 6, 7, 11, 11, 16, 18, 26, 29, 34, 42, 51, 62, 72, 84, 101, 119, 142, 166, 191, 226, 262, 300, 354, 405, 467, 540, 623, 705, 807, 927, 1060, 1206, 1369, 1551, 1760, 1998, 2248, 2556, 2861, 3236, 3628, 4100, 4587, 5152, 5756 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c. LINKS Table of n, a(n) for n=0..56. EXAMPLE The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12): 21 . . 42 421 431 63 532 542 84 643 653 A5 321 521 432 541 632 642 742 743 843 621 631 821 651 841 752 942 721 5321 921 A21 761 C21 4321 5421 5431 842 6432 6321 6421 B21 6531 7321 5432 7431 6431 7521 6521 8421 7421 9321 8321 54321 MATHEMATICA Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Select[Tuples[#, 3], #[[1]]+#[[2]]==#[[3]]&]!={}&]], {n, 0, 30}] PROG (Python) from itertools import combinations_with_replacement from collections import Counter from sympy.utilities.iterables import partitions def A363226(n): return sum(1 for p in partitions(n) if max(p.values(), default=0)==1 and any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()), 3))) # Chai Wah Wu, Sep 20 2023 CROSSREFS For subsets of {1..n} we have A093971 (sum-full sets), complement A007865. The non-strict version is A363225, ranks A364348 (complement A364347). The complement is counted by A364346, non-strict A364345. A000041 counts partitions, strict A000009. A008284 counts partitions by length, strict A008289. A236912 counts sum-free partitions not re-using parts, complement A237113. A323092 counts double-free partitions, ranks A320340. Cf. A002865, A025065, A026905, A085489, A108917, A237667, A237668 A240861, A275972, A320347, A326083. Sequence in context: A076492 A127462 A106436 * A075758 A125596 A351962 Adjacent sequences: A363223 A363224 A363225 * A363227 A363228 A363229 KEYWORD nonn AUTHOR Gus Wiseman, Jul 19 2023 EXTENSIONS a(31)-a(56) from Chai Wah Wu, Sep 20 2023 STATUS approved

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Last modified May 28 01:12 EDT 2024. Contains 372900 sequences. (Running on oeis4.)