A224956 Number of partitions of n where the difference between consecutive parts is at most 2.

1, 1, 2, 3, 5, 6, 9, 11, 16, 19, 26, 31, 42, 50, 65, 78, 100, 119, 149, 178, 222, 263, 322, 382, 465, 549, 660, 778, 932, 1093, 1299, 1520, 1798, 2096, 2464, 2868, 3357, 3892, 4536, 5247, 6096, 7028, 8133, 9357, 10795, 12388, 14244, 16309, 18706, 21367, 24440, 27857, 31788, 36157

Offset: 0

Author

Joerg Arndt, Apr 21 2013

Keywords

• nonn

Comments

Also (by taking the conjugate), a(n) is the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most twice. - Geoffrey Critzer, Sep 30 2013

Examples

			The a(7)=11 such partitions of 7 are
01:  [ 1 1 1 1 1 1 1 ]
02:  [ 2 1 1 1 1 1 ]
03:  [ 2 2 1 1 1 ]
04:  [ 2 2 2 1 ]
05:  [ 3 1 1 1 1 ]
06:  [ 3 2 1 1 ]
07:  [ 3 2 2 ]
08:  [ 3 3 1 ]
09:  [ 4 2 1 ]
10:  [ 4 3 ]
11:  [ 7 ]
The a(7)=11 partitions with no part (excepting the largest) repeated more than twice are the conjugates of the above respectively:
01:  [7]
02:  [6 1]
03:  [5 2]
04:  [4 3]
05:  [5 1 1]
06:  [4 2 1]
07:  [3 3 1]
08:  [3 2 2]
09:  [3 2 1 1]
10:  [2 2 2 1]
11:  [1 1 1 1 1 1 1]
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, `if`(j>0, 0, 1)), j=t..n/i)))
    end:
a:= n-> add(b(n, k, 1), k=0..n):
seq(a(n), n=0..70);  #  Alois P. Heinz, May 01 2013

Mathematica

nn=53;CoefficientList[Series[1+Sum[x^k/(1-x^k)Product[1+x^i+x^(2i),{i,1,k-1}],{k,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 30 2013 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, If[j>0, 0, 1]], {j, t, n/i}]]]; a[n_] := Sum[b[n, k, 1], {k, 0, n}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Jun 19 2015, after Alois P. Heinz *)

PARI

N=66;  q = 'q + O('q^N);
Vec ( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, 1+q^i+q^(2*i) ) ) )
\\ Joerg Arndt, Mar 08 2014

Formula

O.g.f.: 1 + sum(k>=1, x^k/(1-x^k) * prod(i=1..k-1, 1+x^i+x^(2*i) ) ). - Geoffrey Critzer, Sep 30 2013

a(n) = Sum_{k=0..2} A238353(n,k). - Alois P. Heinz, Mar 09 2014

a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022