A351008 - OEIS

A351008

Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093

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OFFSET

0,6

LINKS

Table of n, a(n) for n=0..55.

FORMULA

Conjecture: a(n+1) = A122129(n+1) - A122130(n). - Gus Wiseman, Feb 21 2022

EXAMPLE

The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):

21 31 32 42 43 53 54 64 65 75 76

41 51 52 62 63 73 74 84 85

61 71 72 82 83 93 94

3221 81 91 92 A2 A3

4221 4321 A1 B1 B2

5221 4331 4332 C1

5321 5331 5332

6221 5421 5431

6321 6331

7221 6421

7321

8221

MATHEMATICA

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

CROSSREFS

The version for equal instead of unequal is A035363.

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

This is the even-length case of A122129, opposite A122135.

The odd-length version appears to be A122130.

The alternately unequal and equal version is A351007, any length A351006.

Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.

Sequence in context: A092833 A280664 A100926 * A258875 A179241 A316082

Adjacent sequences: A351005 A351006 A351007 * A351009 A351010 A351011

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 31 2022

STATUS

approved