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A218703
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Number of partitions of n in which any two distinct parts differ by at least 8.
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2
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1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 4, 10, 7, 12, 13, 17, 16, 23, 21, 30, 30, 34, 35, 47, 43, 51, 52, 66, 63, 81, 77, 100, 99, 120, 121, 156, 150, 185, 189, 234, 230, 283, 281, 343, 350, 409, 414, 503, 497, 587, 600, 695, 703, 824, 830, 967, 988, 1122, 1148, 1333
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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 8 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^(8*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(-8*x)) dx = -1.1447921975208768146551512630331558734964408879... - Vaclav Kotesovec, Jan 28 2022
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EXAMPLE
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a(9) = 3: [1,1,1,1,1,1,1,1,1], [3,3,3], [9].
a(10) = 5: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,9], [10].
a(11) = 4: [1,1,1,1,1,1,1,1,1,1,1], [1,1,9], [1,10], [11].
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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-8), 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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