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Number of compositions of n where each part after the first is either twice, half, or equal to the prior part.
14

%I #17 Jun 10 2021 20:36:23

%S 1,1,2,4,6,9,17,24,41,67,109,173,296,469,781,1284,2109,3450,5713,9349,

%T 15422,25351,41720,68590,112982,185753,305752,503041,827819,1361940,

%U 2241435,3687742,6068537,9985389,16431144,27036576,44489533,73205429,120460062,198214516,326161107

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

%H Alois P. Heinz, <a href="/A342340/b342340.txt">Table of n, a(n) for n = 0..4623</a> (first 1001 terms from Andrew Howroyd)

%e The a(1) = 1 through a(6) = 17 compositions:

%e (1) (2) (3) (4) (5) (6)

%e (11) (12) (22) (122) (24)

%e (21) (112) (212) (33)

%e (111) (121) (221) (42)

%e (211) (1112) (222)

%e (1111) (1121) (1122)

%e (1211) (1212)

%e (2111) (1221)

%e (11111) (2112)

%e (2121)

%e (2211)

%e (11112)

%e (11121)

%e (11211)

%e (12111)

%e (21111)

%e (111111)

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

%p b(n-j, j), j=`if`(i=0, {$1..n}, select(x->

%p x::integer and x<=n, {i/2, i, 2*i}))))

%p end:

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

%p seq(a(n), n=0..42); # _Alois P. Heinz_, May 24 2021

%t 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}]

%t (* Second program: *)

%t 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 &]]}]];

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

%t a /@ Range[0, 42] (* _Jean-François Alcover_, Jun 10 2021, after _Alois P. Heinz_ *)

%o (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

%Y The case of partitions is A342337.

%Y The anti-run version is A342331.

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

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

%Y A154402 counts partitions with adjacent parts x = 2y.

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

%Y A274199 counts compositions with adjacent parts x < 2y.

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

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

%Y A342098 counts partitions with adjacent parts x > 2y.

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

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

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

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

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

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 12 2021

%E Terms a(21) and beyond from _Andrew Howroyd_, Mar 13 2021