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A327262
a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 4.
7
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.
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
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Apr 30 2020
STATUS
approved