login
A342336
Number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y = 2x.
14
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, 123, 144, 176, 200, 232, 268, 311, 363, 412, 485, 577, 658, 743, 875, 999, 1126, 1338, 1562, 1767, 2034, 2365, 2691, 3088, 3596, 4152, 4785, 5479, 6310, 7273, 8304, 9573, 11136, 12799, 14619, 16910, 19425, 22142, 25579
OFFSET
0,4
COMMENTS
Also the number of compositions of n with all adjacent parts (x, y) satisfying x = 2y or y > 2x.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 121 terms from David A. Corneth)
David A. Corneth, PARI program
EXAMPLE
The a(1) = 1 through a(12) = 12 compositions (A = 10, B = 11, C = 12):
1 2 3 4 5 6 7 8 9 A B C
21 13 14 15 16 17 18 19 1A 1B
42 25 26 27 28 29 2A
213 142 215 63 37 38 39
214 1421 216 163 137 84
421 2142 217 218 138
4213 263 219
21421 425 426
4214 1425
14213 2163
4215
14214
MAPLE
b:= proc(n, x) option remember; `if`(n=0, 1, add(
`if`(x=0 or x>2*y or y=2*x, b(n-y, y), 0), y=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Mar 14 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_, x_] := b[n, x] = If[n == 0, 1, Sum[
If[x == 0 || x > 2y || y == 2x, b[n-y, y], 0], {y, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)
PROG
(PARI) \\ See PARI link. David A. Corneth, Mar 12 2021
(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 A274199, or A342098 for partitions.
The second condition alone gives A154402 for partitions.
The case of equality is A342331.
The version allowing equality (i.e., non-strict relations) is A342335.
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).
A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
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.
Sequence in context: A231544 A086420 A328106 * A320908 A356692 A361404
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Mar 10 2021
EXTENSIONS
More terms from Joerg Arndt, Mar 12 2021
STATUS
approved