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Number of compositions of n such that the minimal distance between two identical parts equals two.
3

%I #13 Dec 21 2020 07:16:24

%S 0,0,0,0,1,2,3,8,16,34,57,113,213,396,733,1333,2419,4400,7934,14321,

%T 25687,45947,82085,146410,260547,463021,821669,1456296,2578051,

%U 4559972,8057373,14225124,25096606,44246087,77958821,137283534,241626535,425079358,747501363

%N Number of compositions of n such that the minimal distance between two identical parts equals two.

%H Alois P. Heinz, <a href="/A261984/b261984.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) ~ A003242(n). - _Vaclav Kotesovec_, Sep 08 2015

%e a(4) = 1: 121.

%e a(5) = 2: 131, 212.

%e a(6) = 3: 141, 1212, 2121.

%e a(7) = 8: 151, 232, 313, 1213, 1312, 2131, 3121, 12121.

%e a(8) = 16: 161, 242, 323, 1214, 1232, 1313, 1412, 2123, 2141, 2321, 3131, 3212, 4121, 12131, 13121, 21212.

%p g:= proc(n, i) option remember; `if`(n=0, 1, add(

%p `if`(i=j, 0, g(n-j, j)), j=1..n))

%p end:

%p b:= proc(n, i, m) option remember; `if`(n=0, 0, add(

%p `if`(i=j, 0, `if`(j=m, g(n-j, j), b(n-j, j, i))), j=1..n))

%p end:

%p a:= n-> b(n, 0$2):

%p seq(a(n), n=0..45);

%t g[n_, i_] := g[n, i] = If[n==0, 1, Sum[If[i==j, 0, g[n-j, j]], {j, 1, n}]];

%t b[n_, i_, m_] := b[n, i, m] = If[n==0, 0, Sum[If[i==j, 0, If[j==m, g[n-j, j], b[n-j, j, i]]], {j, 1, n}]];

%t a[n_] := b[n, 0, 0];

%t a /@ Range[0, 45] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A261981.

%K nonn

%O 0,6

%A _Alois P. Heinz_, Sep 07 2015