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 A224956 Number of partitions of n where the difference between consecutive parts is at most 2. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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 Alois P. Heinz, Table of n, a(n) for n = 0..1000 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 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 A060714 A241819 Adjacent sequences:  A224953 A224954 A224955 * A224957 A224958 A224959 KEYWORD nonn AUTHOR Joerg Arndt, Apr 21 2013 STATUS approved

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Last modified July 25 03:53 EDT 2021. Contains 346283 sequences. (Running on oeis4.)