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%I #8 May 24 2021 06:41:42
%S 1,1,1,3,1,3,5,5,3,11,9,11,17,15,29,39,31,39,65,57,107,127,149,155,
%T 187,265,293,419,523,571,781,763,941,1371,1387,2125,2383,2775,3243,
%U 4189,4555,5349,7241,7997,10591,13171,14581,17213,20253,25177,27701,34317
%N Number of strict compositions of n with all adjacent parts (x, y) satisfying x <= 2y and y <= 2x.
%C Each quotient of adjacent parts is between 1/2 and 2 inclusive.
%e The a(1) = 1 through a(12) = 17 strict compositions (A = 10, B = 11, C = 12):
%e 1 2 3 4 5 6 7 8 9 A B C
%e 12 23 24 34 35 36 46 47 48
%e 21 32 42 43 53 45 64 56 57
%e 123 124 54 235 65 75
%e 321 421 63 532 74 84
%e 234 1234 236 246
%e 243 1243 245 345
%e 324 3421 542 354
%e 342 4321 632 435
%e 423 1235 453
%e 432 5321 534
%e 543
%e 642
%e 1236
%e 1245
%e 5421
%e 6321
%t Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],And@@Table[#[[i]]<=2*#[[i-1]]&&#[[i-1]]<=2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]
%Y The non-strict version is A224957.
%Y The case with strict relations is A342341 (non-strict: A342330).
%Y A000929 counts partitions with adjacent parts x >= 2y.
%Y A002843 counts compositions with adjacent parts x <= 2y.
%Y A154402 counts partitions with adjacent parts x = 2y.
%Y A274199 counts compositions with adjacent parts x < 2y.
%Y A342094 counts partitions with adjacent x <= 2y (strict: A342095).
%Y A342096 counts partitions without adjacent x >= 2y (strict: A342097).
%Y A342098 counts partitions with adjacent parts x > 2y.
%Y A342331 counts compositions with adjacent parts x = 2y or y = 2x.
%Y A342332 counts compositions with adjacent parts x > 2y or y > 2x.
%Y A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
%Y A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
%Y A342337 counts partitions with adjacent parts x = y or x = 2y.
%Y A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
%Y Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342340.
%K nonn
%O 0,4
%A _Gus Wiseman_, Mar 12 2021
%E a(40)-a(51) from _Alois P. Heinz_, May 24 2021
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