A237666 -id:A237666 - cp's OEIS frontend

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

Clark Kimberling, Feb 11 2014

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

			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.)

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

Cf. A034296, A237666, A092265.

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

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 *)

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

A366154 Irregular triangle read by rows: T(n,k) is the number of integer partitions of n with at least one part a_i such that a_i - a_{i+k} = k.

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 1, 7, 3, 1, 11, 3, 2, 15, 7, 3, 1, 22, 9, 4, 2, 30, 15, 7, 4, 1, 42, 20, 11, 6, 2, 56, 32, 16, 9, 4, 1, 77, 40, 22, 12, 7, 2, 101, 61, 33, 19, 11, 4, 1, 135, 78, 44, 26, 16, 7, 2, 176, 112, 61, 39, 23, 12, 4, 1, 231, 142, 81, 52, 32, 18, 7, 2

Offset: 0

John Tyler Rascoe, Oct 01 2023

Empirical: The first k terms of each column are A000070, for columns k > 0.

			Triangle begins:
      k=0   1  2  3  4
  n=0:  0
  n=1:  1
  n=2:  2
  n=3:  3,  1
  n=4:  5,  1
  n=5:  7,  3, 1
  n=6: 11,  3, 2
  n=7: 15,  7, 3, 1
  n=8: 22,  9, 4, 2
  n=9: 30, 15, 7, 4, 1
  ...
T(7,1) = 7: T(7,2) = 3: T(7,3) = 1:
      (43)        (331)      (4111)
     (421)       (3211)
     (322)      (31111)
    (3211)
    (2221)
   (22111)
  (211111)

John Tyler Rascoe, Python program

Cf. A000041 (column k=0), A237666 (column k=1).

Cf. A000070, A008615, A268193, A354234.

Python
```
# see linked program
```

cp's OEIS Frontend

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

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