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

0,3

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

Links

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

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

Example

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]

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 *)

Prog

(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

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).

Sequence in context: A115270 A339277 A027588 * A131995 A363066 A060714

Adjacent sequences: A224953 A224954 A224955 * A224957 A224958 A224959

Keyword

nonn

Author

Joerg Arndt, Apr 21 2013

Status

approved