|
%I #10 Mar 20 2021 13:49:58
%S 1,1,1,1,1,3,1,3,3,5,5,5,9,7,13,15,17,19,29,31,39,43,63,59,75,121,119,
%T 169,167,199,279,305,343,479,537,733,789,883,1057,1421,1545,1831,2409,
%U 2577,3343,4001,4657,5131,6065,7755,8841,10473,12995,14659,17671,20619,25157,28255,33131,38265,47699,53171,62611,80005,88519,105937,119989
%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 exclusive.
%H Bert Dobbelaere, <a href="/A342341/b342341.txt">Table of n, a(n) for n = 0..100</a>
%e The a(1) = 1 through a(17) = 17 compositions (A..G = 10..16):
%e 1 2 3 4 5 6 7 8 9 A B C D E F G
%e 23 34 35 45 46 47 57 58 59 69 6A
%e 32 43 53 54 64 56 75 67 68 78 79
%e 234 235 65 345 76 86 87 97
%e 432 532 74 354 85 95 96 A6
%e 435 346 347 357 358
%e 453 643 356 456 457
%e 534 653 465 475
%e 543 743 546 547
%e 2345 564 574
%e 2354 645 745
%e 4532 654 754
%e 5432 753 853
%e 2346 2347
%e 6432 2356
%e 6532
%e 7432
%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 unordered version (partitions) is A342097 (non-strict: A342096).
%Y The non-strict version is A342330.
%Y The version allowing equality is A342342 (non-strict: A224957).
%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 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, A342339, A342340.
%K nonn
%O 0,6
%A _Gus Wiseman_, Mar 12 2021
%E More terms from _Bert Dobbelaere_, Mar 19 2021
|
