A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

1, 2, 4, 6, 11, 15, 27, 43, 68, 116, 189, 311, 519, 860, 1433, 2380, 3968, 6613, 11018, 18374, 30633, 51089, 85208, 142113, 237055, 395409, 659576, 1100262, 1835382, 3061711, 5107445, 8520122, 14213135, 23710173, 39553138, 65982316, 110071459, 183620990, 306316328

Offset: 1

Author

Gus Wiseman, May 12 2019

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n.

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

Examples

			The a(1) = 1 through a(6) = 15 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (22)    (23)     (33)
             (21)   (112)   (32)     (222)
             (111)  (121)   (122)    (1122)
                    (211)   (212)    (1212)
                    (1111)  (221)    (1221)
                            (1112)   (2112)
                            (1121)   (2121)
                            (1211)   (2211)
                            (2111)   (11112)
                            (11111)  (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Row sums of A309931.

Cf. A000079, A008965, A034297, A173258, A325351, A325551, A325553, A325558, A325589, A325590.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[DeleteCases[Differences[Append[#,First[#]]],0]]&]],{n,15}]

PARI

step(R,n,D)={matrix(n, n, i, j, if(i>j, sum(k=1, #D, my(s=D[k]); if(j>s && j+s<=n, R[i-j, j-s]))) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)<=1), t=0); while(R, t+=R[n,k]; R=step(R,n,[0,1,-1])); t)} \\ Andrew Howroyd, Aug 23 2019

Formula

a(n) ~ c * d^n, where d = 1.66820206701846111636107... (see A034297), c = 0.65837031047271348106444... - Vaclav Kotesovec, Sep 21 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 23 2019