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

1,2

Comments

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

Links

Table of n, a(n) for n=1..70.

Formula

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

Example

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.

Mathematica

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

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).

Sequence in context: A225607 A227987 A261863 * A344370 A143482 A193679

Adjacent sequences: A327259 A327260 A327261 * A327263 A327264 A327265

Keyword

nonn

Author

Omar E. Pol, Apr 30 2020

Status

approved