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A342333
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Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y >= 2x.
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15
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1, 1, 1, 3, 4, 5, 10, 18, 26, 42, 72, 114, 184, 305, 494, 799, 1305, 2123, 3446, 5611, 9134, 14851, 24162, 39314, 63945, 104025, 169238, 275305, 447863, 728592, 1185248, 1928143, 3136706, 5102743, 8301086, 13504175, 21968436, 35737995, 58138282, 94578751, 153859673
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OFFSET
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0,4
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COMMENTS
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Either quotient x/y or y/x must be >= 2.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(7) = 18 compositions:
(1) (2) (3) (4) (5) (6) (7)
(12) (13) (14) (15) (16)
(21) (31) (41) (24) (25)
(121) (131) (42) (52)
(212) (51) (61)
(141) (124)
(213) (142)
(312) (151)
(1212) (214)
(2121) (241)
(313)
(412)
(421)
(1213)
(1312)
(2131)
(3121)
(12121)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j), j=
`if`(i=0, 1..n, {$1..min(n, iquo(i, 2)), $(2*i)..n})))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Table[#[[i]]>=2*#[[i-1]]||#[[i-1]]>=2*#[[i]], {i, 2, Length[#]}]&]], {n, 0, 15}]
(* Second program: *)
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]}]]];
a[n_] := b[n, 0];
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CROSSREFS
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The unordered version (partitions) is A000929.
Reversing operators and changing 'or' into 'and' gives A224957 (strict: A342342).
The version not allowing equality (i.e., strict relations) is A342332.
The version allowing partial equality is A342334.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
A342098 counts partitions with adjacent parts x > 2y.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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