A327262 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 4.

1, 2, 3, 4, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 30, 32, 17, 54, 19, 40, 42, 44, 23, 72, 25, 52, 54, 84, 29, 90, 31, 96, 66, 68, 35, 144, 37, 76, 78, 120, 41, 126, 43, 132, 135, 92, 47, 192, 49, 150, 102, 156, 53, 162, 110, 168, 114, 116, 59, 300, 61, 124, 126, 192, 130, 264, 67, 204, 138, 210

Offset: 1

Author

Omar E. Pol, Apr 30 2020

Keywords

• nonn

Comments

The one-part partition n = n is included in the count.

Examples

			For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. The sum of the parts is [28] + [16 + 12] + [13 + 9 + 5 + 1] = 84, so a(28) = 84.
		

Crossrefs

Cf. A334460, A334461.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), this sequence (k=4), A334733 (k=5).

Programs

Mathematica

pn4[n_]:=Total[Flatten[Select[IntegerPartitions[n],Union[Abs[Differences[#]]]=={4}&]]]+n; Array[pn4,70] (* Harvey P. Dale, Nov 26 2023 *)

Formula

a(n) = n*A334461(n).