%I #20 Jun 03 2021 13:44:12

%S 1,1,2,3,4,4,7,6,9,10,12,11,19,14,20,24,27,24,37,31,44,45,49,48,71,61,

%T 72,80,92,84,118,102,128,132,144,151,191,166,197,211,244,226,287,263,

%U 313,330,348,347,435,399,462,476,524,508,614,591,674,680,732,731,890,814,916,966,1042,1032,1188,1135,1280,1303

%N Number of integer partitions of n with all adjacent parts (x, y) satisfying either x = y or x = 2y.

%H Alois P. Heinz, <a href="/A342337/b342337.txt">Table of n, a(n) for n = 0..10000</a>

%e The a(1) = 1 through a(9) = 10 partitions:

%e 1 2 3 4 5 6 7 8 9

%e 11 21 22 221 33 421 44 63

%e 111 211 2111 42 2221 422 333

%e 1111 11111 222 22111 2222 4221

%e 2211 211111 4211 22221

%e 21111 1111111 22211 42111

%e 111111 221111 222111

%e 2111111 2211111

%e 11111111 21111111

%e 111111111

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

%p j=`if`(i=0, 1..n, select(x-> x<=n, [i, 2*i]))))

%p end:

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

%p seq(a(n), n=0..80); # _Alois P. Heinz_, May 24 2021

%t Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i-1]]||#[[i-1]]==2*#[[i]],{i,2,Length[#]}]&]],{n,0,30}]

%t (* Second program: *)

%t b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j],

%t {j, If[i == 0, Range[n], Select[{i, 2i}, # <= n&]]}]];

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

%t a /@ Range[0, 80] (* _Jean-François Alcover_, Jun 03 2021, after _Alois P. Heinz_ *)

%Y The first condition alone gives A000005 (for partitions).

%Y The second condition alone gives A154402 (for partitions).

%Y The Heinz numbers of these partitions are given by A342339.

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

%Y A342094 counts partitions with adjacent parts 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 A342331 counts compositions with adjacent parts x = 2y or y = 2x.

%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 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. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342340.

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 10 2021