A342336 - OEIS

%I #39 May 09 2021 12:44:36

%S 1,1,1,2,2,2,4,6,5,6,8,10,12,15,19,22,25,28,37,41,46,62,72,79,95,113,

%T 123,144,176,200,232,268,311,363,412,485,577,658,743,875,999,1126,

%U 1338,1562,1767,2034,2365,2691,3088,3596,4152,4785,5479,6310,7273,8304,9573,11136,12799,14619,16910,19425,22142,25579

%N Number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y = 2x.

%C Also the number of compositions of n with all adjacent parts (x, y) satisfying x = 2y or y > 2x.

%H Alois P. Heinz, <a href="/A342336/b342336.txt">Table of n, a(n) for n = 0..5000</a> (first 121 terms from David A. Corneth)

%H David A. Corneth, <a href="/A342336/a342336.gp.txt">PARI program</a>

%e The a(1) = 1 through a(12) = 12 compositions (A = 10, B = 11, C = 12):

%e   1   2   3    4    5    6     7     8      9      A       B       C

%e           21   13   14   15    16    17     18     19      1A      1B

%e                          42    25    26     27     28      29      2A

%e                          213   142   215    63     37      38      39

%e                                214   1421   216    163     137     84

%e                                421          2142   217     218     138

%e                                                    4213    263     219

%e                                                    21421   425     426

%e                                                            4214    1425

%e                                                            14213   2163

%e                                                                    4215

%e                                                                    14214

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

%p      `if`(x=0 or x>2*y or y=2*x, b(n-y, y), 0), y=1..n))

%p     end:

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

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Mar 14 2021

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]>2*#[[i-1]]||#[[i-1]]==2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]

%t (* Second program: *)

%t b[n_, x_] := b[n, x] = If[n == 0, 1, Sum[

%t      If[x == 0 || x > 2y || y == 2x, b[n-y, y], 0], {y, 1, n}]];

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

%t a /@ Range[0, 80] (* _Jean-François Alcover_, May 09 2021, after _Alois P. Heinz_ *)

%o (PARI) See PARI link \\ _David A. Corneth_, Mar 12 2021

%o (PARI)

%o C(n, pred)={my(M=matid(n)); for(k=1, n, for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); sum(q=1, n, M[q, ])}

%o seq(n)={concat([1], C(n, (i,j)->i>2*j || j==2*i))} \\ _Andrew Howroyd_, Mar 13 2021

%Y The first condition alone gives A274199, or A342098 for partitions.

%Y The second condition alone gives A154402 for partitions.

%Y The case of equality is A342331.

%Y The version allowing equality (i.e., non-strict relations) is A342335.

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

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

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

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

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

%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 A342337 counts partitions with adjacent parts x = y or x = 2y.

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

%Y A342342 counts strict compositions with adjacent parts x <= 2y and y <= 2x.

%Y Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342340.

%K nonn

%O 0,4

%A _Gus Wiseman_, Mar 10 2021

%E More terms from _Joerg Arndt_, Mar 12 2021