A256321 - OEIS

%I #10 Oct 29 2017 12:48:07

%S 0,2,8,19,33,52,75,102,133,169,208,252,300,352,408,469,533,602,675,

%T 752,833,919,1008,1102,1200,1302,1408,1519,1633,1752,1875,2002,2133,

%U 2269,2408,2552,2700,2852,3008,3169,3333,3502,3675,3852,4033,4219,4408,4602

%N Number of partitions of 5n into exactly 3 parts.

%H Colin Barker, <a href="/A256321/b256321.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

%F a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6) for n>5.

%F G.f.: -x*(x^2+2*x+2)*(2*x^2+2*x+1) / ((x-1)^3*(x+1)*(x^2+x+1)).

%e For n=1 the 2 partitions of 5*1 = 5 are [1, 1, 3] and [1, 2, 2].

%t Length /@ (Total /@ IntegerPartitions[5 #, {3}] & /@ Range[0, 47]) (* _Michael De Vlieger_, Mar 24 2015 *)

%t LinearRecurrence[{1,1,0,-1,-1,1},{0,2,8,19,33,52},50] (* _Harvey P. Dale_, Oct 29 2017 *)

%o (PARI) concat(0, vector(40, n, k=0; forpart(p=5*n, k++, , [3,3]); k))

%o (PARI) concat(0, Vec(-x*(x^2+2*x+2)*(2*x^2+2*x+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^100)))

%Y Cf. A033428 (6n), A256320 (4n), A256322 (7n).

%K nonn,easy

%O 0,2

%A _Colin Barker_, Mar 24 2015