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Number of partitions of n in which any two distinct parts differ by at least 6.
2

%I #12 Jan 28 2022 15:47:03

%S 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,

%T 85,106,106,141,138,174,180,219,224,284,282,342,356,422,431,530,532,

%U 631,660,765,789,948,965,1123,1184,1356,1408,1658,1703,1967,2076

%N Number of partitions of n in which any two distinct parts differ by at least 6.

%C Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 6 times.

%H Vaclav Kotesovec, <a href="/A218701/b218701.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%F G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(6*i)/(1-x^i)).

%F 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

%e a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].

%e a(7) = 2: [1,1,1,1,1,1,1], [7].

%e a(8) = 5: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,7], [8].

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p b(n, i-1) +add(b(n-i*j, i-6), j=1..n/i)))

%p end:

%p a:= n-> b(n, n):

%p seq(a(n), n=0..70);

%Y Column k=6 of A218698.

%Y Cf. A160976.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 04 2012