A229362 a(n) = n for n = 1, 2, 3; for n > 3: a(n) = number of partitions of n into preceding terms.

1, 2, 3, 4, 6, 10, 12, 17, 21, 29, 34, 47, 55, 71, 84, 107, 124, 156, 180, 221, 256, 310, 355, 428, 488, 578, 660, 775, 879, 1027, 1160, 1342, 1516, 1743, 1958, 2243, 2513, 2858, 3198, 3621, 4037, 4556, 5065, 5689, 6316, 7069, 7824, 8733, 9644, 10726, 11827

Offset: 1

Author

Reinhard Zumkeller, Sep 21 2013

Keywords

• nonn

Examples

			a(4) = #{3+1, 2+2, 2+1+1, 1+1+1+1} = 4 < A000041(4) = 5;
a(5) = #{4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 5x1} = 6 < A000041(5) = 7;
a(6) = #{6, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+4x1, 6x1} = 10 < A000041(6) = 11;
a(7) = #{6+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+4x1, 2+2+2+1, 2+2+1+1+1, 2+5x1, 7x1} = 12 < A000041(7) = 15.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A007279.

Programs

Haskell

a229362 n = a229362_list !! (n-1)
a229362_list = 1 : 2 : 3 : f 4 [1,2,3] where
   f x ys = y : f (x + 1) (ys ++ [y]) where y = p ys x
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

Mathematica

a[n_] := a[n] = If[n<4, n, IntegerPartitions[n, All, Array[a, n-1]] // Length];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 12 2019 *)