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A335443 Number of compositions of n where neighboring runs have different lengths.

%I A335443 #26 Apr 13 2022 13:02:03
%S A335443 1,1,2,2,5,8,13,24,42,68,122,210,360,622,1077,1858,3198,5519,9549,
%T A335443 16460,28386,49031,84595,145988,251956,434805,750418,1294998,2234971,
%U A335443 3857106,6656383,11487641,19825318,34214136,59046458,101901743,175860875,303498779
%N A335443 Number of compositions of n where neighboring runs have different lengths.
%H A335443 Alois P. Heinz, <a href="/A335443/b335443.txt">Table of n, a(n) for n = 0..1000</a>
%e A335443 a(0) = 1: the empty composition.
%e A335443 a(1) = 1: 1.
%e A335443 a(2) = 2: 2, 11.
%e A335443 a(3) = 2: 3, 111.
%e A335443 a(4) = 5: 4, 22, 112, 211, 1111.
%e A335443 a(5) = 8: 5, 113, 122, 221, 311, 1112, 2111, 11111.
%e A335443 a(6) = 13: 6, 33, 114, 222, 411, 1113, 1221, 2112, 3111, 11112, 11211, 21111, 111111.
%e A335443 a(7) = 24: 7, 115, 133, 223, 322, 331, 511, 1114, 1222, 2113, 2221, 3112, 4111, 11113, 11122, 11311, 21112, 22111, 31111, 111112, 111211, 112111, 211111, 1111111.
%e A335443 a(8) = 42: 8, 44, 116, 224, 233, 332, 422, 611, 1115, 1223, 1331, 2114, 2222, 3113, 3221, 4112, 5111, 11114, 11222, 11411, 12221, 21113, 22211, 31112, 41111, 111113, 111122, 111221, 111311, 112112, 113111, 122111, 211112, 211211, 221111, 311111, 1111112, 1111211, 1112111, 1121111, 2111111, 11111111.
%p A335443 b:= proc(n, l, t) option remember; `if`(n=0, 1, add(add(
%p A335443       `if`(j=t, 0, b(n-i*j, i, j)), j=1..n/i), i={$1..n} minus {l}))
%p A335443     end:
%p A335443 a:= n-> b(n, 0$2):
%p A335443 seq(a(n), n=0..40);
%t A335443 b[n_, l_, t_] := b[n, l, t] = If[n == 0, 1, Sum[Sum[If[j == t, 0,
%t A335443      b[n-i*j, i, j]], {j, 1, n/i}], {i, Range[n]~Complement~{l}}]];
%t A335443 a[n_] := b[n, 0, 0];
%t A335443 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 13 2022, after _Alois P. Heinz_ *)
%Y A335443 Cf. A003242, A011782, A329738, A329739, A329748, A329749, A329766, A335942.
%K A335443 nonn
%O A335443 0,3
%A A335443 _Alois P. Heinz_, Jul 06 2020