A237665 Number of partitions of n such that the distinct terms arranged in increasing order form a string of two or more consecutive integers.

0, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 17, 24, 27, 35, 39, 50, 57, 70, 79, 97, 111, 132, 150, 178, 204, 239, 271, 316, 361, 416, 472, 545, 618, 706, 800, 912, 1032, 1173, 1320, 1496, 1687, 1902, 2137, 2410, 2702, 3034, 3398, 3808, 4258, 4765, 5313, 5932, 6613

Offset: 0

Author

Clark Kimberling, Feb 11 2014

Keywords

• nonn
• easy

Comments

Number of partitions of n with maximal distance between parts = 1; column k=1 of A238353. [Joerg Arndt, Mar 23 2014]

Conjecture: a(n+1) = sum of smallest parts in the distinct partitions of n with an even number of parts. - George Beck, May 06 2017

Examples

			The qualifying partitions of 8 are 332, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 6.  (The strings of distinct parts arranged in increasing order are 23, 123, 123, 12, 12, 12.)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Shane Chern (Xiaohang Chen), On a conjecture of George Beck. II, 2018.

Crossrefs

Cf. A034296, A237666, A092265.

Programs

Maple

b:= proc(n, i, t) option remember;
      `if`(n=0 or i=1, `if`(n=0 and t=2 or n>0 and t>0, 1, 0),
      `if`(i>n, 0, add(b(n-i*j, i-1, min(t+1, 2)), j=1..n/i)))
    end:
a:= n-> add(b(n, i, 0), i=1..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 15 2014

Mathematica

Map[Length[Select[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], (Table[-1, {Length[#]}] == # && # =!= \{}) &]] &, Range[55]] (* Peter J. C. Moses, Feb 09 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0 || i==1, If[n==0 && t==2 || n>0 && t > 0, 1, 0], If[i>n, 0, Sum[b[n-i*j, i-1, Min[t+1, 2]], {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Formula

a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jan 28 2022