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%I #8 Jan 27 2022 20:47:01
%S 0,0,0,1,1,2,4,5,8,12,18,25,36,48,65,89,119,157,207,269,350,448,574,
%T 729,927,1166,1465,1830,2282,2827,3501,4309,5300,6483,7923,9641,11718,
%U 14187,17155,20674,24885,29860,35787,42772,51054,60791,72289,85772,101641
%N Number of integer partitions of n with at least two adjacent parts of quotient 2.
%e The a(3) = 1 through a(9) = 12 partitions:
%e (21) (211) (221) (42) (421) (422) (63)
%e (2111) (321) (2221) (521) (621)
%e (2211) (3211) (3221) (3321)
%e (21111) (22111) (4211) (4221)
%e (211111) (22211) (5211)
%e (32111) (22221)
%e (221111) (32211)
%e (2111111) (42111)
%e (222111)
%e (321111)
%e (2211111)
%e (21111111)
%t Table[Length[Select[IntegerPartitions[n], MemberQ[Divide@@@Partition[#,2,1],2]&]],{n,0,30}]
%Y The complement is counted by A350837, strict A350840.
%Y The complimentary additive version is A350842, strict A350844.
%Y These partitions are ranked by A350845, complement A350838.
%Y A000041 = integer partitions.
%Y A323092 = double-free integer partitions, ranked by A320340.
%Y Cf. A000929, A003000, A003114, A018819, A045690, A045691, A116931, A120641, A154402, A323093, A342094, A342095, A342096, A342098.
%K nonn
%O 0,6
%A _Gus Wiseman_, Jan 20 2022
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