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A218701
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Number of partitions of n in which any two distinct parts differ by at least 6.
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2
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1, 1, 2, 2, 3, 2, 4, 2, 5, 5, 8, 7, 14, 11, 16, 19, 23, 22, 32, 29, 38, 40, 48, 48, 67, 63, 81, 85, 106, 106, 141, 138, 174, 180, 219, 224, 284, 282, 342, 356, 422, 431, 530, 532, 631, 660, 765, 789, 948, 965, 1123, 1184, 1356, 1408, 1658, 1703, 1967, 2076
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OFFSET
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0,3
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COMMENTS
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Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 6 times.
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LINKS
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FORMULA
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G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(6*i)/(1-x^i)).
log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-6*x)) dx = -1.0551351195231385243969621008395374852115209081... - Vaclav Kotesovec, Jan 28 2022
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EXAMPLE
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a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
a(7) = 2: [1,1,1,1,1,1,1], [7].
a(8) = 5: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,7], [8].
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-6), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..70);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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