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A238424
Number of partitions of n without three consecutive parts in arithmetic progression.
31
1, 1, 2, 2, 4, 5, 6, 8, 13, 13, 19, 24, 30, 36, 47, 54, 72, 85, 106, 123, 151, 178, 220, 256, 314, 362, 432, 505, 605, 692, 827, 953, 1121, 1303, 1522, 1729, 2037, 2321, 2691, 3095, 3577, 4061, 4699, 5334, 6126, 6959, 7966, 9005, 10317, 11638, 13252, 14977
OFFSET
0,3
COMMENTS
Also the number of partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences. - Gus Wiseman, Mar 31 2020
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..300 from Joerg Arndt and Alois P. Heinz, terms 301..350 from Fausto A. C. Cariboni)
EXAMPLE
The a(8) = 13 such partitions are:
01: [ 3 2 2 1 ]
02: [ 3 3 1 1 ]
03: [ 3 3 2 ]
04: [ 4 2 1 1 ]
05: [ 4 2 2 ]
06: [ 4 3 1 ]
07: [ 4 4 ]
08: [ 5 2 1 ]
09: [ 5 3 ]
10: [ 6 1 1 ]
11: [ 6 2 ]
12: [ 7 1 ]
13: [ 8 ]
MATHEMATICA
a[n_, r_, d_] := a[n, r, d] = Block[{j}, If[n == 0, 1, Sum[If[j == r+d, 0, a[n-j, j, j - r]], {j, Min[n, r]}]]]; a[n_] := a[n, 2*n+1, 0]; a /@ Range[0, 100] (* Giovanni Resta, Mar 02 2014 *)
Table[Length[Select[IntegerPartitions[n], !MemberQ[Differences[#, 2], 0]&]], {n, 0, 30}] (* Gus Wiseman, Mar 31 2020 *)
CROSSREFS
Cf. A238433 (partitions avoiding equidistant arithmetic progressions).
Cf. A238571 (partitions avoiding any arithmetic progression).
Cf. A238687.
The version for compositions is A238423, with strict case A325849.
The version for permutations is A295370.
The strict case is A332668.
The Heinz numbers of these partitions are the complement of A333195.
Partitions with equal differences are A049988.
Sequence in context: A293957 A238687 A238433 * A121269 A211860 A250114
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 26 2014
STATUS
approved