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A342333 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y >= 2x.

%I A342333 #21 Jun 10 2021 20:35:39
%S A342333 1,1,1,3,4,5,10,18,26,42,72,114,184,305,494,799,1305,2123,3446,5611,
%T A342333 9134,14851,24162,39314,63945,104025,169238,275305,447863,728592,
%U A342333 1185248,1928143,3136706,5102743,8301086,13504175,21968436,35737995,58138282,94578751,153859673
%N A342333 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y >= 2x.
%C A342333 Either quotient x/y or y/x must be >= 2.
%H A342333 Alois P. Heinz, <a href="/A342333/b342333.txt">Table of n, a(n) for n = 0..2000</a>
%e A342333 The a(1) =  1 through a(7) = 18 compositions:
%e A342333   (1)  (2)  (3)   (4)    (5)    (6)     (7)
%e A342333             (12)  (13)   (14)   (15)    (16)
%e A342333             (21)  (31)   (41)   (24)    (25)
%e A342333                   (121)  (131)  (42)    (52)
%e A342333                          (212)  (51)    (61)
%e A342333                                 (141)   (124)
%e A342333                                 (213)   (142)
%e A342333                                 (312)   (151)
%e A342333                                 (1212)  (214)
%e A342333                                 (2121)  (241)
%e A342333                                         (313)
%e A342333                                         (412)
%e A342333                                         (421)
%e A342333                                         (1213)
%e A342333                                         (1312)
%e A342333                                         (2131)
%e A342333                                         (3121)
%e A342333                                         (12121)
%p A342333 b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j), j=
%p A342333       `if`(i=0, 1..n, {$1..min(n, iquo(i, 2)), $(2*i)..n})))
%p A342333     end:
%p A342333 a:= n-> b(n, 0):
%p A342333 seq(a(n), n=0..42);  # _Alois P. Heinz_, May 24 2021
%t A342333 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]>=2*#[[i-1]]||#[[i-1]]>=2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]
%t A342333 (* Second program: *)
%t A342333 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j], {j, 1, n}], Sum[b[n-j, j], {j, Range[Min[n, Quotient[i, 2]]]~Union~Range[2i, n]}]]];
%t A342333 a[n_] := b[n, 0];
%t A342333 a /@ Range[0, 42] (* _Jean-François Alcover_, Jun 10 2021, after_Alois P. Heinz_ *)
%Y A342333 The unordered version (partitions) is A000929.
%Y A342333 Reversing operators and changing 'or' into 'and' gives A224957 (strict: A342342).
%Y A342333 The version not allowing equality (i.e., strict relations) is A342332.
%Y A342333 The version allowing partial equality is A342334.
%Y A342333 A002843 counts compositions with adjacent parts x <= 2y.
%Y A342333 A154402 counts partitions with adjacent parts x = 2y.
%Y A342333 A274199 counts compositions with adjacent parts x < 2y.
%Y A342333 A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
%Y A342333 A342096 counts partitions without adjacent x >= 2y (strict: A342097).
%Y A342333 A342098 counts partitions with adjacent parts x > 2y.
%Y A342333 A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
%Y A342333 A342331 counts compositions with adjacent parts x = 2y or y = 2x.
%Y A342333 A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
%Y A342333 A342337 counts partitions with adjacent parts x = y or x = 2y.
%Y A342333 A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
%Y A342333 Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342340.
%K A342333 nonn
%O A342333 0,4
%A A342333 _Gus Wiseman_, Mar 10 2021
%E A342333 More terms from _Joerg Arndt_, Mar 12 2021