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A002573 Restricted partitions.
(Formerly M1062 N0399)
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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]

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.

EXAMPLE

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)

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) ];

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 *)

CROSSREFS

Cf. A002572, A047913, A002574, A049284, A049285.

A column of the triangle in A176431.

Sequence in context: A257932 A026713 A254685 * A064492 A000072 A268306

Adjacent sequences:  A002570 A002571 A002572 * A002574 A002575 A002576

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.