A256320 Number of partitions of 4n into exactly 3 parts.

0, 1, 5, 12, 21, 33, 48, 65, 85, 108, 133, 161, 192, 225, 261, 300, 341, 385, 432, 481, 533, 588, 645, 705, 768, 833, 901, 972, 1045, 1121, 1200, 1281, 1365, 1452, 1541, 1633, 1728, 1825, 1925, 2028, 2133, 2241, 2352, 2465, 2581, 2700, 2821, 2945, 3072, 3201

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

a(n) = A184637(n) for n > 2.

a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5) for n>4.

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

a(n) = 2*(6*n^2+cos((2*Pi*n)/3)-1)/9. - Colin Barker, Jun 06 2016

Example

For n=2 the 5 partitions of 4*2 = 8 are [1,1,6], [1,2,5], [1,3,4], [2,2,4] and [2,3,3].

Mathematica

Length /@ (Total /@ IntegerPartitions[4 #, {3}] & /@ Range[0, 49]) (* Michael De Vlieger, Mar 24 2015 *)
CoefficientList[Series[-x (x + 1)^3/((x - 1)^3 (x^2 + x + 1)), {x, 0, 49}], x] (* or *)
Table[2 (6 n^2 + Cos[(2 Pi n)/3] - 1)/9, {n, 0, 49}] (* Michael De Vlieger, Jun 06 2016 *)

Prog

(PARI) concat(0, vector(40, n, k=0; forpart(p=4*n, k++, , [3, 3]); k))
(PARI) concat(0, Vec(-x*(x+1)^3/((x-1)^3*(x^2+x+1)) + O(x^100)))

Crossrefs

Cf. A033428 (6n), A256321 (5n), A256322 (7n).

Cf. A184637.

Sequence in context: A028347 A346379 A354399 * A301693 A038794 A225284

Adjacent sequences: A256317 A256318 A256319 * A256321 A256322 A256323

Keyword

nonn,easy

Author

Colin Barker, Mar 24 2015

Status

approved