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A342338
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Number of compositions of n with all adjacent parts (x, y) satisfying x < 2y and y <= 2x.
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14
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1, 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 73, 106, 155, 224, 328, 477, 695, 1013, 1477, 2154, 3140, 4578, 6673, 9728, 14176, 20663, 30113, 43882, 63940, 93167, 135747, 197776, 288138, 419773, 611522, 890829, 1297685, 1890305, 2753505, 4010804, 5842113, 8509462
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OFFSET
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0,3
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COMMENTS
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Also the number of compositions of n with all adjacent parts (x, y) satisfying x <= 2y and y < 2x.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(7) = 12 compositions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (23) (33) (34)
(111) (211) (32) (42) (43)
(1111) (221) (222) (223)
(2111) (321) (232)
(11111) (2211) (322)
(21111) (421)
(111111) (2221)
(3211)
(22111)
(211111)
(1111111)
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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: *)
c[n_, pred_] := Module[{M = IdentityMatrix[n], i, k}, For[k = 1, k <= n, k++, For[i = 1, i <= k - 1, i++, M[[i, k]] = Sum[If[pred[j, i], M[[j, k - i]], 0], {j, 1, k - i}]]]; Sum[M[[q, All]], {q, 1, n}]];
pred[i_, j_] := i < 2j && j <= 2i;
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PROG
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(PARI)
C(n, pred)={my(M=matid(n)); for(k=1, n, for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); sum(q=1, n, M[q, ])}
seq(n)={concat([1], C(n, (i, j)->i<2*j && j<=2*i))} \\ Andrew Howroyd, Mar 13 2021
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CROSSREFS
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The first condition alone gives A274199.
The second condition alone gives A002843.
Reversing operators and changing 'and' to 'or' gives A342334.
The version with both relations strict is A342341.
The version with neither relation strict is A342342.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A224957 counts compositions with x <= 2y and y <= 2x.
A274199 counts compositions with adjacent parts x < 2y.
A342098 counts partitions with adjacent parts x > 2y.
A342330 counts compositions with x < 2y and y < 2x.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 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.
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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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