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A128508 Number of partitions p of n such that max(p) - min(p) = 3.

%I A128508 #23 Jan 13 2025 17:50:44
%S A128508 0,0,0,0,0,1,1,3,3,7,7,12,14,20,22,32,34,45,51,63,69,87,93,112,124,
%T A128508 144,156,184,196,225,245,275,295,335,355,396,426,468,498,552,582,637,
%U A128508 679,735,777,847,889,960,1016,1088,1144,1232,1288,1377,1449,1539,1611,1719
%N A128508 Number of partitions p of n such that max(p) - min(p) = 3.
%C A128508 See A008805 and A049820 for the numbers of partitions p of n such that max(p)-min(p)=1 or 2, respectively.
%H A128508 Alois P. Heinz, <a href="/A128508/b128508.txt">Table of n, a(n) for n = 0..1000</a>
%H A128508 G. E. Andrews, M. Beck and N. Robbins, <a href="http://arxiv.org/abs/1406.3374">Partitions with fixed differences between largest and smallest parts</a>, arXiv:1406.3374 [math.NT], 2014
%F A128508 Conjecture. a(1)=0 and, for n>1, a(n+1)=a(n)+d(n), where d(n) is defined as follows: d=0,0,0,1,0 for n=1,...,5 and, for n>5, d(n)=d(n-2)+1 if n=6k or n=6k+4, d(n)=d(n-2) if n=6k+1 or n=6k+3, d(n)=d(n-2)+2Floor[n/6] if n=6k+2 and d(n)=d(n-5) if n=6k+5.
%F A128508 G.f. for number of partitions p of n such that max(p)-min(p) = m is Sum_{k>0} x^(2*k+m)/Product_{i=0..m} (1-x^(k+i)). - _Vladeta Jovovic_, Jul 04 2007
%F A128508 a(n) = A097364(n,3) = A116685(n,3) = A117143(n) - A117142(n). - _Alois P. Heinz_, Nov 02 2012
%t A128508 np[n_]:=Length[Select[IntegerPartitions[n],Max[#]-Min[#]==3&]]; Array[np,60] (* _Harvey P. Dale_, Jul 02 2012 *)
%Y A128508 Cf. A008805, A049820, A097364, A116685, A117142, A117143.
%K A128508 nonn
%O A128508 0,8
%A A128508 _John W. Layman_, May 07 2007
%E A128508 More terms from _Vladeta Jovovic_, Jul 04 2007