A214248 -id:A214248 - cp's OEIS frontend

A214255 Number of compositions of n where differences between neighboring parts are in {-2,...,2}.

1, 1, 2, 4, 8, 14, 27, 49, 92, 170, 317, 587, 1097, 2038, 3798, 7072, 13176, 24538, 45720, 85166, 158670, 295596, 550708, 1025974, 1911445, 3561079, 6634457, 12360279, 23027789, 42901825, 79928175, 148909982, 277426505, 516858952, 962933307, 1793991419

Offset: 0

Alois P. Heinz, Jul 08 2012

nonn

			a(3) = 4: [3], [2,1], [1,2], [1,1,1].
a(4) = 8: [4], [3,1], [2,2], [2,1,1], [1,3], [1,2,1], [1,1,2], [1,1,1,1].
a(5) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=2 of A214248.

b:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
      `if`(n=i, 1, add(b(n-i, i+j), j=-2..2)))
    end:
a:= n-> `if`(n=0, 1, add(b(n, j), j=1..n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n-i, i+j], {j, -2, 2}]]]; a[n_] := If[n == 0, 1, Sum[b[n, j], {j, 1, n}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2014, after Alois P. Heinz *)

a(n) ~ c * d^n, where d = 1.8630486786572002290749607226537419966705160765891889162715127426..., c = 0.6251341184281574379681933375704862852528326365321195333127800734... . - Vaclav Kotesovec, Sep 02 2014

A368557 Number of compositions of n such that the set of absolute differences is a subset of the set of parts.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 11, 10, 13, 27, 58, 87, 157, 253, 438, 850, 1462, 2474, 4472, 7716, 13544, 24115, 42360, 74013, 131038, 229009, 401946, 707293, 1242059, 2177682, 3828831, 6716062, 11777179, 20678592, 36267148, 63586772, 111556751, 195610763, 342949281

Offset: 0

John Tyler Rascoe, Dec 29 2023

nonn

			For n=12, composition [2,1,2,4,3] of 12 has the set of absolute differences {1,2}, which is a subset of the set of parts {1,2,3,4}, so it counts towards a(12) = 157.
a(3) = 3 compositions: [3], [2,1], [1,2].
a(6) = 11 compositions: [6], [4,2], [2,4], [3,2,1], [3,1,2], [2,3,1], [2,1,3], [1,3,2], [1,2,3], [2,1,2,1], [1,2,1,2].

John Tyler Rascoe, Python program

Cf. A003242, A032020, A173258, A214248, A214270.

g[0] = {{}}; g[n_Integer] := g[n] = Flatten[ParallelTable[Append[#, i] & /@ g[n - i], {i, 1, n}], 1];
isC[p_List] := Module[{d}, d = Abs[Differences[p]]; Union[d] === Union[Select[d, MemberQ[p, #] &]]];
a[n_Integer] := a[n] = Count[g[n], p_ /; isC[p]];
Monitor[Table[a[n], {n, 0, 19}], {n, Table[a[m], {m, 0, n - 1}]}] (* Robert P. P. McKone, Jan 02 2024 *)

a(24)-a(38) from Alois P. Heinz, Dec 30 2023

cp's OEIS Frontend

A214255 Number of compositions of n where differences between neighboring parts are in {-2,...,2}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula