A224956 - OEIS

%I #40 Jan 26 2022 05:13:45

%S 1,1,2,3,5,6,9,11,16,19,26,31,42,50,65,78,100,119,149,178,222,263,322,

%T 382,465,549,660,778,932,1093,1299,1520,1798,2096,2464,2868,3357,3892,

%U 4536,5247,6096,7028,8133,9357,10795,12388,14244,16309,18706,21367,24440,27857,31788,36157

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

%C 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

%H Vaclav Kotesovec, <a href="/A224956/b224956.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%F 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

%F a(n) = Sum_{k=0..2} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014

%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * n^(3/4)). - _Vaclav Kotesovec_, Jan 26 2022

%e The a(7)=11 such partitions of 7 are

%e 01:  [ 1 1 1 1 1 1 1 ]

%e 02:  [ 2 1 1 1 1 1 ]

%e 03:  [ 2 2 1 1 1 ]

%e 04:  [ 2 2 2 1 ]

%e 05:  [ 3 1 1 1 1 ]

%e 06:  [ 3 2 1 1 ]

%e 07:  [ 3 2 2 ]

%e 08:  [ 3 3 1 ]

%e 09:  [ 4 2 1 ]

%e 10:  [ 4 3 ]

%e 11:  [ 7 ]

%e The a(7)=11 partitions with no part (excepting the largest) repeated more than twice are the conjugates of the above respectively:

%e 01:  [7]

%e 02:  [6 1]

%e 03:  [5 2]

%e 04:  [4 3]

%e 05:  [5 1 1]

%e 06:  [4 2 1]

%e 07:  [3 3 1]

%e 08:  [3 2 2]

%e 09:  [3 2 1 1]

%e 10:  [2 2 2 1]

%e 11:  [1 1 1 1 1 1 1]

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

%p       add(b(n-i*j, i-1, `if`(j>0, 0, 1)), j=t..n/i)))

%p     end:

%p a:= n-> add(b(n, k, 1), k=0..n):

%p seq(a(n), n=0..70);  #  _Alois P. Heinz_, May 01 2013

%t 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 *)

%t 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_ *)

%o (PARI)

%o N=66;  q = 'q + O('q^N);

%o Vec ( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, 1+q^i+q^(2*i) ) ) )

%o \\ _Joerg Arndt_, Mar 08 2014

%Y 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).

%K nonn

%O 0,3

%A _Joerg Arndt_, Apr 21 2013