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b[n_, i_] := b[n, i] = If[n == 0, 0, Sum[If[i == j, Ceiling[2^(n-j-1)], b[n-j, j]], {j, 1, n}]];

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

Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz's Maple code *)

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From Gus Wiseman, Jul 07 2020: (Start)

The a(2) = 1 through a(6) = 18 compositions:

(1,1) (1,1,1) (2,2) (1,1,3) (3,3)

(1,1,2) (1,2,2) (1,1,4)

(2,1,1) (2,2,1) (2,2,2)

(1,1,1,1) (3,1,1) (4,1,1)

(1,1,1,2) (1,1,1,3)

(1,1,2,1) (1,1,2,2)

(1,2,1,1) (1,1,3,1)

(2,1,1,1) (1,2,2,1)

(1,1,1,1,1) (1,3,1,1)

(2,1,1,2)

(2,2,1,1)

(3,1,1,1)

(1,1,1,1,2)

(1,1,1,2,1)

(1,1,2,1,1)

(1,2,1,1,1)

(2,1,1,1,1)

(1,1,1,1,1,1)

(End)

a(n) is the sumSum of positive-indexed terms of row n of A106356.

Compositions where each part is adjacent to an equal part are A114901.

Cf. A114901, A178470, A242882, A244164, A325534, A335448, A335452.

Cf. A008965, A059966, A178470, A181796, A242882, A244164, A329738, A335448, A335452.

Compositions with equal multiplicities are A098504.

Compositions with equal runsparts togethercontiguous are A274174.

Permutations of prime indices with runs together are A335448.

Cf. A008965, A059966, A178470, A181796, A242882, A244164, A329738, A335448, A335452.