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A175342
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Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.
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43
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1, 2, 4, 5, 6, 10, 8, 10, 15, 14, 12, 22, 14, 18, 28, 21, 18, 34, 20, 28, 38, 28, 24, 46, 31, 32, 48, 38, 30, 62, 32, 40, 58, 42, 46, 73, 38, 46, 68, 58, 42, 84, 44, 56, 90, 56, 48, 94, 55, 70, 90, 66, 54, 106, 70, 74, 100, 70, 60, 130, 62, 74, 118, 81, 82, 130, 68, 84, 120
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: x/(1-x) + Sum_{k>=2} x^k * (1 + x^(k(k-1)/2)) / (1 - x^(k(k-1)/2)) / (1 -x^k).
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EXAMPLE
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The a(1) = 1 through a(8) = 10 compositions with equal differences:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (12) (13) (14) (15) (16) (17)
(21) (22) (23) (24) (25) (26)
(111) (31) (32) (33) (34) (35)
(1111) (41) (42) (43) (44)
(11111) (51) (52) (53)
(123) (61) (62)
(222) (1111111) (71)
(321) (2222)
(111111) (11111111)
(End)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], SameQ@@Differences[#]&]], {n, 0, 15}] (* returns a(0) = 1, Gus Wiseman, May 15 2019*)
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CROSSREFS
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Cf. A000005, A000079, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548, A325557, A325558.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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