A002573 Restricted partitions.
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
Offset: 1
Examples
From _Joerg Arndt_, Dec 18 2012: (Start) 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)
References
- 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).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..500
- Shimon Even & Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.
- H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
Programs
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Maple
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
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 *)
Comments