A342330 Number of compositions of n with all adjacent parts (x,y) satisfying x < 2y and y < 2x.

1, 1, 2, 2, 3, 4, 4, 7, 9, 11, 17, 23, 32, 44, 63, 91, 127, 180, 255, 363, 516, 732, 1044, 1485, 2109, 3002, 4277, 6089, 8660, 12323, 17550, 24986, 35562, 50628, 72084, 102616, 146077, 207980, 296114, 421555, 600153, 854469, 1216543, 1731983, 2465842, 3510713

Offset: 0

Author

Gus Wiseman, Mar 09 2021

Keywords

• nonn

Comments

Each quotient of adjacent parts is between 1/2 and 2 exclusive.

Examples

			The a(1) = 1 through a(9) = 11 partitions:
  1   2    3     4      5       6        7         8          9
      11   111   22     23      33       34        35         45
                 1111   32      222      43        44         54
                        11111   111111   223       53         234
                                         232       233        333
                                         322       323        432
                                         1111111   332        2223
                                                   2222       2232
                                                   11111111   2322
                                                              3222
                                                              111111111
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Andrew Howroyd)

Crossrefs

The version allowing equality is A224957.

The unordered version (partitions) is A342096, with strict case A342097.

Reversing operators and changing 'and' into 'or' gives A342332.

The version allowing partial equality is A342338.

The strict case is A342341.

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

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

A154402 counts partitions with all adjacent parts x = 2y.

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

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

A342098 counts partitions with all adjacent parts x > 2y.

A342331 counts compositions where each part is twice or half the prior.

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

A342337 counts compositions with all adjacent parts x = y or x = 2y.

Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342333, A342334, A342336, A342339, A342340.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j)
      , j=`if`(i=0, 1..n, floor(i/2)+1..min(n, 2*i-1))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 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, 1, Floor[i/2] + 1], If[i == 0, n, Min[n, 2i - 1]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 45] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

PARI

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, ])}
seq(n)={concat([1], C(n, (i,j)->i<2*j && j<2*i))} \\ Andrew Howroyd, Mar 13 2021

Extensions

Terms a(31) and beyond from Andrew Howroyd, Mar 13 2021