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

0,5

Links

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

Example

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.

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)<n, 0,
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j=0, 1, x) , j=0..2)))))
    end:
a:= n-> (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 *)

Crossrefs

Column k=1 of A262191.

Sequence in context: A141290 A010752 A049929 * A280308 A289121 A060738

Adjacent sequences: A262189 A262190 A262191 * A262193 A262194 A262195

Keyword

nonn

Author

Alois P. Heinz, Sep 14 2015

Status

approved