A237666 Number of partitions of n that include a pair of consecutive integers.

0, 0, 0, 1, 1, 3, 3, 7, 9, 15, 20, 32, 40, 61, 78, 112, 142, 199, 250, 341, 428, 568, 710, 930, 1151, 1486, 1835, 2334, 2868, 3615, 4413, 5513, 6706, 8298, 10052, 12359, 14895, 18195, 21857, 26526, 31747, 38337, 45702, 54923, 65272, 78062, 92481, 110168, 130089

Offset: 0

Author

Clark Kimberling, Feb 11 2014

Keywords

• nonn
• easy

Examples

			The qualifying partitions of 8 are 521, 431, 332, 421, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 9.
		

Links

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

Crossrefs

Cf. A116931, A237665.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i, l) option remember; `if`(n=0 or i<1, 0,
       b(n, i-1, 0) +add(`if`(i+1=l, g(n-i*j, i-1),
       b(n-i*j, i-1, i)), j=1..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 14 2014

Mathematica

Map[Length[Cases[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], {_, -1, _}]] &, Range[50]]  (* Peter J. C. Moses, Feb 09 2014 *)
g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_, l_] := b[n, i, l] = If[n==0 || i<1, 0, b[n, i-1, 0] + Sum[If[i+1 == l, g[n-i*j, i-1], b[n-i*j, i-1, i]], {j, 1, n/i}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 01 2016, after Alois P. Heinz *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Jan 28 2022

Conjecture: for n > 0, a(n) = A000041(n) - A116931(n). - Vaclav Kotesovec, Jan 28 2022