A351006 - OEIS

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A351006

Number of integer partitions of n into parts that are alternately unequal and equal.

1, 1, 1, 2, 3, 4, 4, 6, 7, 9, 10, 12, 14, 16, 18, 22, 25, 28, 31, 36, 40, 46, 50, 56, 64, 71, 78, 88, 96, 106, 118, 130, 143, 158, 172, 190, 209, 228, 248, 274, 298, 324, 354, 384, 418, 458, 494, 536, 584, 631, 683, 742, 800, 864, 936, 1010, 1088, 1176, 1264

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..58.

EXAMPLE

The a(1) = 1 through a(11) = 12 partitions (A = 10, B = 11):

1 2 3 4 5 6 7 8 9 A B

21 31 32 42 43 53 54 64 65

211 41 51 52 62 63 73 74

311 411 61 71 72 82 83

322 422 81 91 92

511 611 522 433 A1

3221 711 622 533

4221 811 722

32211 5221 911

42211 4331

6221

52211

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]

CROSSREFS

Without equalities we have A122129, opposite A122135, even-length A351008.

The non-strict version is A351003, opposite A351004, even-length A351012.

The alternately equal and unequal version is A351005, even-length A035457.

The even-length case is A351007.

Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A344605, A350842, A350844, A351011.

Sequence in context: A248517 A364031 A007896 * A325682 A241088 A074139

Adjacent sequences: A351003 A351004 A351005 * A351007 A351008 A351009

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 31 2022

STATUS

approved