A342340 Number of compositions of n where each part after the first is either twice, half, or equal to the prior part.

1, 1, 2, 4, 6, 9, 17, 24, 41, 67, 109, 173, 296, 469, 781, 1284, 2109, 3450, 5713, 9349, 15422, 25351, 41720, 68590, 112982, 185753, 305752, 503041, 827819, 1361940, 2241435, 3687742, 6068537, 9985389, 16431144, 27036576, 44489533, 73205429, 120460062, 198214516, 326161107

Offset: 0

Author

Gus Wiseman, Mar 12 2021

Keywords

• nonn

Examples

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (22)    (122)    (24)
             (21)   (112)   (212)    (33)
             (111)  (121)   (221)    (42)
                    (211)   (1112)   (222)
                    (1111)  (1121)   (1122)
                            (1211)   (1212)
                            (2111)   (1221)
                            (11111)  (2112)
                                     (2121)
                                     (2211)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4623 (first 1001 terms from Andrew Howroyd)

Crossrefs

The case of partitions is A342337.

The anti-run version is A342331.

A000929 counts partitions with adjacent parts x >= 2y.

A002843 counts compositions with adjacent parts x <= 2y.

A154402 counts partitions with adjacent parts x = 2y.

A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).

A274199 counts compositions with adjacent parts x < 2y.

A342094 counts partitions with adjacent x <= 2y (strict: A342095).

A342096 counts partitions without adjacent x >= 2y (strict: A342097).

A342098 counts partitions with adjacent parts x > 2y.

A342330 counts compositions with x < 2y and y < 2x (strict: A342341).

A342332 counts compositions with adjacent parts x > 2y or y > 2x.

A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.

A342334 counts compositions with adjacent parts x >= 2y or y > 2x.

A342335 counts compositions with adjacent parts x >= 2y or y = 2x.

A342338 counts compositions with adjacent parts x < 2y and y <= 2x.

Cf. A000005, A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342339.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-j, j), j=`if`(i=0, {$1..n}, select(x->
       x::integer and x<=n, {i/2, i, 2*i}))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, May 24 2021

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i-1]]||#[[i]]==2*#[[i-1]]||#[[i-1]]==2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j], {j, If[i == 0, Range[n], Select[ {i/2, i, 2 i}, IntegerQ[#] && # <= n &]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 42] (* Jean-François Alcover, Jun 10 2021, after Alois P. Heinz *)

PARI

seq(n)={my(M=matid(n)); for(k=1, n, for(i=1, k-1, M[i, k] = if(i%2==0, M[i/2,k-i]) + if(i*2<=k, M[i,k-i]) + if(i*3<=k, M[i*2,k-i]))); concat([1], sum(q=1, n, M[q, ]))} \\ Andrew Howroyd, Mar 13 2021

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 13 2021