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 A262192 Number of compositions of n such that the maximal distance between two identical parts equals one. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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) (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

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Last modified May 18 10:28 EDT 2022. Contains 353807 sequences. (Running on oeis4.)