A262192 Number of compositions of n such that the maximal distance between two identical parts equals one.

0, 0, 1, 0, 3, 4, 5, 12, 21, 36, 43, 88, 133, 222, 331, 450, 753, 1120, 1703, 2508, 3753, 5010, 7807, 11020, 16243, 22974, 33277, 46764, 63639, 91822, 127943, 180048, 249585, 348204, 480361, 664618, 884833, 1237470, 1675087, 2299104, 3103203, 4234072, 5700371

Offset: 0

Author

Alois P. Heinz, Sep 14 2015

Keywords

• nonn

Examples

			a(2) = 1: 11.
a(4) = 3: 22, 112, 211.
a(5) = 4: 113, 122, 221, 311.
a(6) = 5: 33, 114, 411, 1122, 2211.
a(7) = 12: 115, 133, 223, 322, 331, 511, 1123, 1132, 2113, 2311, 3112, 3211.
a(8) = 21: 44, 116, 224, 233, 332, 422, 611, 1124, 1133, 1142, 1223, 1322, 2114, 2213, 2231, 2411, 3122, 3221, 3311, 4112, 4211.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..900 (terms 0..650 from Alois P. Heinz)

Crossrefs

Column k=1 of A262191.

Programs

Maple

g:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), g(n-k, k)+k*g(n-k, k-1)))
    end:
b:= proc(n, i) option remember; expand(`if`(i*(i+1) (p-> add(coeff(p, x, i)*i!, i=0..degree(p)))(b(n$2))
        -add(g(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..50);

Mathematica

g[n_, k_] := g[n, k] = If[k < 0 || n < 0, 0, If[k == 0, If[n == 0, 1, 0], g[n - k, k] + k*g[n - k, k - 1]]];
b[n_, i_] := b[n, i] = Expand[If[i(i+1) < n, 0, If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 0, 1, x], {j, 0, 2}]]]]];
a[n_] := With[{p = b[n, n]}, Sum[Coefficient[p, x, i]*i!, {i, 0, Exponent[p, x]}]] - Sum[g[n, k], {k, 0, Floor[(Sqrt[8n + 1] - 1)/2]}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)