A342335 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y = 2x.

1, 1, 1, 3, 3, 3, 7, 9, 9, 16, 21, 22, 36, 47, 51, 77, 101, 114, 165, 217, 251, 350, 459, 540, 733, 962, 1152, 1535, 2010, 2437, 3207, 4192, 5141, 6698, 8728, 10802, 13979, 18170, 22652, 29169, 37814, 47410, 60854, 78716, 99144, 126974, 163897, 207159, 264918, 341331, 432606, 552693, 711013, 903041, 1153060

Offset

0,4

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Example

The a(1) = 1 through a(9) = 16 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)      (8)      (9)
            (12)  (13)   (14)   (15)    (16)     (17)     (18)
            (21)  (121)  (212)  (24)    (25)     (26)     (27)
                                (42)    (124)    (125)    (36)
                                (213)   (142)    (215)    (63)
                                (1212)  (214)    (242)    (126)
                                (2121)  (421)    (1214)   (216)
                                        (1213)   (1421)   (1215)
                                        (12121)  (21212)  (1242)
                                                          (2124)
                                                          (2142)
                                                          (2421)
                                                          (4212)
                                                          (21213)
                                                          (121212)
                                                          (212121)

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: *)
c[n_, pred_] := Module[{M = IdentityMatrix[n], i, k}, For[k = 1, k <= n, k++, For[i = 1, i <= k-1, i++, M[[i, k]] = Sum[If[pred[j, i], M[[j, k-i]], 0], {j, 1, k-i}]]]; Sum[M[[q, All]], {q, 1, n}]];
pred[i_, j_] := i >= 2j || j == 2i;
Join[{1}, c[60, pred]] (* Jean-François Alcover, Jun 10 2021, after Andrew Howroyd *)

Prog

(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

Crossrefs

The first condition alone gives A002843, or A000929 for partitions.

The second condition alone gives A154402 for partitions.

The case of equality is A342331.

The version not allowing equality (i.e., strict relations) is A342336.

A224957 counts compositions with adjacent parts x <= 2y and y <= 2x.

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

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).

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

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

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

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

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

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

Sequence in context: A227826 A268127 A200076 * A137438 A098524 A143015

Adjacent sequences: A342332 A342333 A342334 * A342336 A342337 A342338

Keyword

nonn

Author

Gus Wiseman, Mar 10 2021

Extensions

More terms from Joerg Arndt, Mar 12 2021

Status

approved