A342337 Number of integer partitions of n with all adjacent parts (x, y) satisfying either x = y or x = 2y.

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

Offset: 0

Author

Gus Wiseman, Mar 10 2021

Keywords

• nonn

Examples

			The a(1) = 1 through a(9) = 10 partitions:
  1   2    3     4      5       6        7         8          9
      11   21    22     221     33       421       44         63
           111   211    2111    42       2221      422        333
                 1111   11111   222      22111     2222       4221
                                2211     211111    4211       22221
                                21111    1111111   22211      42111
                                111111             221111     222111
                                                   2111111    2211111
                                                   11111111   21111111
                                                              111111111
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

The first condition alone gives A000005 (for partitions).

The second condition alone gives A154402 (for partitions).

The Heinz numbers of these partitions are given by A342339.

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

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

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

A274199 counts compositions with adjacent parts x < 2y.

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

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

A342098 counts partitions with adjacent parts x > 2y.

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

A342331 counts compositions with adjacent parts x = 2y or y = 2x.

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

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

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

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

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

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j),
      j=`if`(i=0, 1..n, select(x-> x<=n, [i, 2*i]))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 24 2021

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i-1]]||#[[i-1]]==2*#[[i]],{i,2,Length[#]}]&]],{n,0,30}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j],
     {j, If[i == 0, Range[n], Select[{i, 2i}, # <= n&]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 03 2021, after Alois P. Heinz *)