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

0,6

Links

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

Example

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

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 *)

Crossrefs

Cf. A116931, A237665.

Sequence in context: A331788 A206433 A301589 * A285187 A034411 A258289

Adjacent sequences: A237663 A237664 A237665 * A237667 A237668 A237669

Keyword

nonn,easy

Author

Clark Kimberling, Feb 11 2014

Status

approved