A293569 Partitions with designated summands in which no parts are multiples of 3.
1, 1, 3, 4, 9, 12, 21, 29, 48, 64, 99, 132, 195, 257, 366, 480, 666, 864, 1173, 1511, 2016, 2576, 3384, 4296, 5574, 7027, 9015, 11296, 14355, 17880, 22527, 27908, 34896, 43008, 53406, 65508, 80844, 98711, 121128, 147272, 179784, 217704, 264489, 319064
Offset: 0
Keywords
Examples
n = 3 n = 4 n = 5
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2'+ 1' 4' 5'
1'+ 1 + 1 2'+ 2 4'+ 1'
1 + 1'+ 1 2 + 2' 2'+ 2 + 1'
1 + 1 + 1' 2'+ 1'+ 1 2 + 2'+ 1'
2'+ 1 + 1' 2'+ 1'+ 1 + 1
1'+ 1 + 1 + 1 2'+ 1 + 1'+ 1
1 + 1'+ 1 + 1 2'+ 1 + 1 + 1'
1 + 1 + 1'+ 1 1'+ 1 + 1 + 1 + 1
1 + 1 + 1 + 1' 1 + 1'+ 1 + 1 + 1
1 + 1 + 1'+ 1 + 1
1 + 1 + 1 + 1'+ 1
1 + 1 + 1 + 1 + 1'
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a(3) = 4. a(4) = 9. a(5) = 12.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
Programs
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Mathematica
nmax = 50; CoefficientList[Series[Product[(1-x^(6*k))^2 / ( (1-x^k)^2 * (1+x^k) * (1+x^(9*k)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2017 *) -
Ruby
def partition(n, min, max) return [[]] if n == 0 [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}} end def A(k, n) partition(n, 1, n).select{|i| i.all?{|j| j % k > 0}}.map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values.inject(:*)}.inject(:+) end def A293569(n) [1] + (1..n).map{|i| A(3, i)} end p A293569(40)
Formula
Expansion of eta(q^6)^2 * eta(q^9) / (eta(q) * eta(q^2) * eta(q^18)) in powers of q.
a(n) ~ 5^(1/4) * exp(2*Pi*sqrt(5*n/3)/3) / (2 * 3^(7/4)* n^(3/4)). - Vaclav Kotesovec, Oct 13 2017