A002574 Restricted partitions.
0, 0, 1, 1, 2, 4, 7, 13, 24, 42, 76, 137, 245, 441, 792, 1420, 2550, 4576, 8209, 14732, 26433, 47424, 85092, 152670, 273914, 491453, 881744, 1581985, 2838333, 5092398, 9136528, 16392311, 29410243, 52766343, 94670652, 169853138, 304741614, 546751437, 980952673, 1759973660
Offset: 1
Examples
From _Joerg Arndt_, Dec 18 2012: (Start) There are a(8)=13 compositions 8=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1): [ 1] [ 3 1 1 1 1 1 ] [ 2] [ 3 1 1 1 2 ] [ 3] [ 3 1 1 2 1 ] [ 4] [ 3 1 2 1 1 ] [ 5] [ 3 1 2 2 ] [ 6] [ 3 2 1 1 1 ] [ 7] [ 3 2 1 2 ] [ 8] [ 3 2 2 1 ] [ 9] [ 3 2 3 ] [10] [ 3 3 1 1 ] [11] [ 3 3 2 ] [12] [ 3 4 1 ] [13] [ 3 5 ] (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..200
- Shimon Even and 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(3,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[3, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 05 2013, after Maple *)
Extensions
More terms from Michael Somos
Comments