A173258 Number of compositions of n where differences between neighboring parts are in {-1,1}.

1, 1, 1, 3, 2, 4, 5, 5, 7, 10, 9, 14, 16, 19, 24, 31, 35, 45, 55, 66, 84, 104, 124, 156, 192, 236, 292, 363, 444, 551, 681, 839, 1040, 1287, 1586, 1967, 2430, 3001, 3717, 4597, 5683, 7034, 8697, 10758, 13312, 16469, 20369, 25204, 31180, 38574, 47726, 59047

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

John Tyler Rascoe, Illustration of n = 1..16

Formula

a(n) ~ c * d^n, where d=1.23729141259673487395949649334678514763130846902468..., c=1.134796087242490181499736234755111281606636700030106.... - Vaclav Kotesovec, May 01 2014

G.f.: 1 + Sum_{k>0} G(x,k) where G(x,k) = x^k*(1 + G(x,k+1) + G(x,k-1)) for k > 0 and G(x,0) = 0. - John Tyler Rascoe, Sep 16 2023

Example

a(3) = 3: [3], [2,1], [1,2].
a(4) = 2: [4], [1,2,1].
a(5) = 4: [5], [3,2], [2,3], [2,1,2].
a(6) = 5: [6], [3,2,1], [2,1,2,1], [1,2,3], [1,2,1,2].

Maple

b:= proc(n, i) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j), j=[-1, 1])))
    end:
a:= n-> `if`(n=0, 1, add(b(n, j), j=1..n)):
seq(a(n), n=0..70);

Mathematica

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j], {j, {-1, 1}}]]]; a[n_] := If[n == 0, 1, Sum[b[n, j], {j, 1, n}]]; Table[a[n], {n, 0, 70}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Prog

(PARI)
step(R, n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + if(j+1<=n, R[i-j, j+1])) )}
a(n)={my(R=matid(n), t=(n==0), m=0); while(R, m++; t+=vecsum(R[n, ]); R=step(R, n)); t} \\ Andrew Howroyd, Aug 23 2019

Crossrefs

Column k=1 of A214247, A214249.

Row sums of A309938, A364039.

Cf. A227310, A291904, A291905, A343795, A362500, A363718, A364529.

Sequence in context: A286298 A128440 A063201 * A039858 A035558 A089401

Adjacent sequences: A173255 A173256 A173257 * A173259 A173260 A173261

Keyword

nonn

Author

Alois P. Heinz, Jul 08 2012

Status

approved