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 A342331 Number of compositions of n where each part after the first is either twice or half the prior part. 14
 1, 1, 1, 3, 2, 2, 5, 4, 3, 9, 6, 4, 14, 9, 8, 22, 15, 11, 37, 24, 21, 58, 40, 30, 95, 67, 53, 157, 114, 85, 264, 187, 147, 428, 315, 244, 732, 527, 410, 1207, 892, 681, 2034, 1490, 1155, 3416, 2508, 1927, 5731, 4215, 3259, 9597, 7091, 5454, 16175, 11914, 9194, 27134, 20033, 15425, 45649, 33672, 25967, 76714 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS All quotients of adjacent parts are either 1/2 or 2. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from Andrew Howroyd) EXAMPLE The a(1) =  1 through a(12) = 14 compositions (A = 10, B = 11, C = 12):   1  2  3   4    5    6     7      8      9       A        B        C         12  121  212  24    124    242    36      424      21242    48         21            42    421    21212  63      12124    24212    84                       1212  12121         1242    12421    2121212  363                       2121                2124    42121             2424                                           2421    1212121           4242                                           4212                      121242                                           121212                    124212                                           212121                    212124                                                                     212421                                                                     242121                                                                     421212                                                                     12121212                                                                     21212121 MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j), j=      `if`(i=0, 1..n, select(t-> t::integer and t<=n, {2*i, i/2}))))     end: a:= n-> b(n, 0): seq(a(n), n=0..80);  # Alois P. Heinz, Mar 14 2021 MATHEMATICA Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Table[#[[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[{2i, i/2}, IntegerQ[#] && # <= n &]]}]]; a[n_] := b[n, 0]; a /@ Range[0, 80] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *) PROG (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*3<=k, M[i*2, k-i]))); concat(, sum(q=1, n, M[q, ]))} \\ Andrew Howroyd, Mar 13 2021 CROSSREFS The unordered version (partitions) is A154402. The version allowing equality is A342340. A000929 counts partitions with all adjacent parts x >= 2y. A002843 counts compositions with all adjacent parts x <= 2y. A224957 counts compositions with all x <= 2y and y <= 2x (strict: A342342). A274199 counts compositions with all adjacent parts x < 2y. A342094 counts partitions with all adjacent x <= 2y (strict: A342095). A342096 counts partitions with no adjacent x >= 2y (strict: A342097). A342098 counts partitions with all adjacent parts x > 2y. A342330 counts compositions with all x < 2y and y < 2x (strict: A342341). A342332 counts compositions with all adjacent parts x > 2y or y > 2x. A342333 counts compositions with adjacent parts x >= 2y or y >= 2x. A342335 counts compositions with all adjacent parts x >= 2y or y = 2x. A342337 counts partitions with all adjacent parts x = y or x = 2y. A342338 counts compositions with all adjacent parts x < 2y and y <= 2x. Cf. A003114, A003242, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342339. Sequence in context: A210956 A282161 A205675 * A058608 A112196 A021035 Adjacent sequences:  A342328 A342329 A342330 * A342332 A342333 A342334 KEYWORD nonn AUTHOR Gus Wiseman, Mar 09 2021 EXTENSIONS More terms from Joerg Arndt, Mar 12 2021 STATUS approved

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Last modified July 30 19:33 EDT 2021. Contains 346359 sequences. (Running on oeis4.)