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%I #11 Feb 01 2022 05:12:18
%S 1,2,2,3,3,4,4,6,6,8,9,11,13,17,19,24,29,35,42,51,61,75,90,108,130,
%T 158,189,227,272,325,389,464,553,659,782,929,1102,1306,1545,1824,2153,
%U 2538,2989,3514,4127,4842,5673,6642,7766,9068,10583,12335,14361,16705
%N Number of integer partitions of n with no adjacent parts having quotient >= 2.
%C The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.
%H Fausto A. C. Cariboni, <a href="/A342096/b342096.txt">Table of n, a(n) for n = 1..250</a>
%e The a(1) = 1 through a(10) = 8 partitions:
%e 1 2 3 4 5 6 7 8 9 A
%e 11 111 22 32 33 43 44 54 55
%e 1111 11111 222 322 53 333 64
%e 111111 1111111 332 432 433
%e 2222 3222 532
%e 11111111 111111111 3322
%e 22222
%e 1111111111
%t Table[Length[Select[IntegerPartitions[n],And@@Thread[Differences[-#]<Rest[#]]&]],{n,30}]
%Y The case of equality (all adjacent parts having quotient 2) is A154402.
%Y The multiplicative version is A342083 or A342084.
%Y The version allowing quotients of 2 exactly is A342094.
%Y The strict case allowing quotients of 2 exactly is A342095.
%Y The strict case is A342097.
%Y The reciprocal version is A342098.
%Y A000009 counts strict partitions.
%Y A000929 counts partitions with no adjacent parts having quotient < 2.
%Y A003114 counts partitions with adjacent parts differing by more than 1.
%Y A034296 counts partitions with adjacent parts differing by at most 1.
%Y Cf. A027193, A001055, A001227, A003242, A167606, A342085, A342191.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 02 2021
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