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A307370
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Number of integer partitions of n with 2 distinct parts, none appearing more than twice.
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2
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0, 0, 0, 1, 2, 4, 4, 6, 7, 7, 10, 10, 11, 12, 15, 13, 17, 16, 19, 18, 22, 19, 25, 22, 26, 24, 30, 25, 32, 28, 34, 30, 37, 31, 40, 34, 41, 36, 45, 37, 47, 40, 49, 42, 52, 43, 55, 46, 56, 48, 60, 49, 62, 52, 64, 54, 67, 55, 70, 58, 71, 60, 75, 61, 77, 64, 79, 66
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OFFSET
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0,5
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COMMENTS
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The Heinz numbers of these partitions appear to be given by A296205.
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LINKS
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FORMULA
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G.f.: x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + a(n-5) - a(n-7) - a(n-8) for n>8. (End)
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EXAMPLE
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The a(3) = 1 through a(10) = 10 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(211) (41) (51) (52) (62) (63) (73)
(221) (411) (61) (71) (72) (82)
(311) (2211) (322) (332) (81) (91)
(331) (422) (441) (433)
(511) (611) (522) (442)
(3311) (711) (622)
(811)
(3322)
(4411)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[Union[#]]==2&&Max@@Length/@Split[#]<=2&]], {n, 0, 30}]
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PROG
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(PARI) concat([0, 0, 0], Vec(x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Apr 08 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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