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Number of strict compositions of n with no three consecutive parts in arithmetic progression.
9

%I #5 Jun 02 2019 00:48:39

%S 1,1,1,3,3,5,9,13,19,23,51,57,91,117,179,283,381,531,737,1017,1335,

%T 2259,2745,3983,5289,7367,9413,13155,19461,25129,33997,45633,61225,

%U 80481,107091,137475,205243,253997,345527,447003,604919,768331,1026167,1299227

%N Number of strict compositions of n with no three consecutive parts in arithmetic progression.

%C A composition of n is a finite sequence of positive integers with sum n. a(n) is the number of strict compositions of n with no two of their adjacent first-differences equal, or with no 0's in their second-differences.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>

%e The a(1) = 1 through a(8) = 19 compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (12) (13) (14) (15) (16) (17)

%e (21) (31) (23) (24) (25) (26)

%e (32) (42) (34) (35)

%e (41) (51) (43) (53)

%e (132) (52) (62)

%e (213) (61) (71)

%e (231) (124) (125)

%e (312) (142) (134)

%e (214) (143)

%e (241) (152)

%e (412) (215)

%e (421) (251)

%e (314)

%e (341)

%e (413)

%e (431)

%e (512)

%e (521)

%t Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],!MemberQ[Differences[#,2],0]&]],{n,0,30}]

%Y The non-strict case is A238423.

%Y Cf. A007862, A049988, A175342, A279945, A295370, A325545, A325874, A325875.

%K nonn

%O 0,4

%A _Gus Wiseman_, May 31 2019