0,5

From Edward Early, Jan 10 2009: (Start)

Also the dimension of the n-th degree part of the mod 3 Steenrod algebra.

Also the number of partitions into parts (3^j-1)/2=1+3+3^2+...+3^(j-1) for j>=1. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (terms n = 1..1000 from Reinhard Zumkeller)

More generally, number of partitions of n into distinct parts such that m*u<=v for all pairs (u,v) of parts with u<v is equal to the number of partitions of n into parts of the form (m^k-1)/(m-1), thus g.f. for the number of such partitions is 1/Product_{k>0} (1-x^((m^k-1)/(m-1))). - Vladeta Jovovic, Jan 09 2009

a(10) = #{10, 9+1, 8+2} = 3;

a(11) = #{11, 10+1, 9+2} = 3;

a(12) = #{12, 11+1, 10+2, 9+3} = 4;

a(13) = #{13, 12+1, 11+2, 10+3, 9+3+1} = 5.

From Joerg Arndt, Dec 28 2012: (Start)

The a(33)=17 such partitions of 33 are

[ 1] [ 24 7 2 ]

[ 2] [ 24 8 1 ]

[ 3] [ 25 6 2 ]

[ 4] [ 25 7 1 ]

[ 5] [ 25 8 ]

[ 6] [ 26 6 1 ]

[ 7] [ 26 7 ]

[ 8] [ 27 5 1 ]

[ 9] [ 27 6 ]

[10] [ 28 4 1 ]

[11] [ 28 5 ]

[12] [ 29 3 1 ]

[13] [ 29 4 ]

[14] [ 30 3 ]

[15] [ 31 2 ]

[16] [ 32 1 ]

[17] [ 33 ]

(End)

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(3^i-1)))

end:

a:= n-> b(2*n, 1+ilog[3](2*n)):

seq(a(n), n=0..77); # Alois P. Heinz, Oct 01 2025

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Function[p, If[p > n, 0, b[n-p, i]]][3^i-1]]];

a[n_] := b[2n, Length[IntegerDigits[2n, 3]]];

Table[a[n], {n, 0, 77}] (* Jean-François Alcover, Jan 20 2026, after Alois P. Heinz *)

(Haskell)

a132011 = p [1..] where

p _ 0 = 1

p (k:ks) m = if m < k then 0 else p [3 * k ..] (m - k) + p ks m

-- Reinhard Zumkeller, Oct 10 2013

Cf. A000009, A000929.

Cf. A147583. - Reinhard Zumkeller, Nov 08 2008

Sequence in context: A088004 A070548 A209628 * A363822 A393466 A054893

Adjacent sequences: A132008 A132009 A132010 * A132012 A132013 A132014

Reinhard Zumkeller, Aug 07 2007

a(0)=1 prepended by Alois P. Heinz, Oct 01 2025

approved