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A002573
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Restricted partitions.
(Formerly M1062 N0399)
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9
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0, 1, 1, 2, 4, 7, 12, 22, 39, 70, 126, 225, 404, 725, 1299, 2331, 4182, 7501, 13458, 24145, 43316, 77715, 139430, 250152, 448808, 805222, 1444677, 2591958, 4650342, 8343380, 14969239, 26856992, 48185362, 86451602, 155106844, 278284440, 499283177, 895787396, 1607174300, 2883507098
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OFFSET
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1,4
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COMMENTS
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Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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There are a(8)=22 compositions 8=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1):
[ 1] [ 2 1 1 1 1 1 1 ]
[ 2] [ 2 1 1 1 1 2 ]
[ 3] [ 2 1 1 1 2 1 ]
[ 4] [ 2 1 1 2 1 1 ]
[ 5] [ 2 1 1 2 2 ]
[ 6] [ 2 1 2 1 1 1 ]
[ 7] [ 2 1 2 1 2 ]
[ 8] [ 2 1 2 2 1 ]
[ 9] [ 2 1 2 3 ]
[10] [ 2 2 1 1 1 1 ]
[11] [ 2 2 1 1 2 ]
[12] [ 2 2 1 2 1 ]
[13] [ 2 2 2 1 1 ]
[14] [ 2 2 2 2 ]
[15] [ 2 2 3 1 ]
[16] [ 2 2 4 ]
[17] [ 2 3 1 1 1 ]
[18] [ 2 3 1 2 ]
[19] [ 2 3 2 1 ]
[20] [ 2 3 3 ]
[21] [ 2 4 1 1 ]
[22] [ 2 4 2 ]
(End)
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MAPLE
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v := proc(c, d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i, d-c), i=1..2*c); fi; end; [ seq(v(2, n), n=1..50) ];
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MATHEMATICA
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v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; a[n_] := v[2, n]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Jan 30 2012, after Maple *)
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CROSSREFS
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A column of the triangle in A176431.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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