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

%I A342340 #17 Jun 10 2021 20:36:23
%S A342340 1,1,2,4,6,9,17,24,41,67,109,173,296,469,781,1284,2109,3450,5713,9349,
%T A342340 15422,25351,41720,68590,112982,185753,305752,503041,827819,1361940,
%U A342340 2241435,3687742,6068537,9985389,16431144,27036576,44489533,73205429,120460062,198214516,326161107
%N A342340 Number of compositions of n where each part after the first is either twice, half, or equal to the prior part.
%H A342340 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 A342340 The a(1) = 1 through a(6) = 17 compositions:
%e A342340   (1)  (2)   (3)    (4)     (5)      (6)
%e A342340        (11)  (12)   (22)    (122)    (24)
%e A342340              (21)   (112)   (212)    (33)
%e A342340              (111)  (121)   (221)    (42)
%e A342340                     (211)   (1112)   (222)
%e A342340                     (1111)  (1121)   (1122)
%e A342340                             (1211)   (1212)
%e A342340                             (2111)   (1221)
%e A342340                             (11111)  (2112)
%e A342340                                      (2121)
%e A342340                                      (2211)
%e A342340                                      (11112)
%e A342340                                      (11121)
%e A342340                                      (11211)
%e A342340                                      (12111)
%e A342340                                      (21111)
%e A342340                                      (111111)
%p A342340 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A342340       b(n-j, j), j=`if`(i=0, {$1..n}, select(x->
%p A342340        x::integer and x<=n, {i/2, i, 2*i}))))
%p A342340     end:
%p A342340 a:= n-> b(n, 0):
%p A342340 seq(a(n), n=0..42);  # _Alois P. Heinz_, May 24 2021
%t A342340 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 A342340 (* Second program: *)
%t A342340 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 A342340 a[n_] := b[n, 0];
%t A342340 a /@ Range[0, 42] (* _Jean-François Alcover_, Jun 10 2021, after _Alois P. Heinz_ *)
%o A342340 (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 A342340 The case of partitions is A342337.
%Y A342340 The anti-run version is A342331.
%Y A342340 A000929 counts partitions with adjacent parts x >= 2y.
%Y A342340 A002843 counts compositions with adjacent parts x <= 2y.
%Y A342340 A154402 counts partitions with adjacent parts x = 2y.
%Y A342340 A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
%Y A342340 A274199 counts compositions with adjacent parts x < 2y.
%Y A342340 A342094 counts partitions with adjacent x <= 2y (strict: A342095).
%Y A342340 A342096 counts partitions without adjacent x >= 2y (strict: A342097).
%Y A342340 A342098 counts partitions with adjacent parts x > 2y.
%Y A342340 A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
%Y A342340 A342332 counts compositions with adjacent parts x > 2y or y > 2x.
%Y A342340 A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
%Y A342340 A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
%Y A342340 A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
%Y A342340 A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
%Y A342340 Cf. A000005, A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342339.
%K A342340 nonn
%O A342340 0,3
%A A342340 _Gus Wiseman_, Mar 12 2021
%E A342340 Terms a(21) and beyond from _Andrew Howroyd_, Mar 13 2021