A325589 Number of compositions of n whose circular differences are all 1 or -1.

0, 0, 2, 0, 2, 2, 2, 4, 4, 2, 8, 6, 8, 10, 12, 16, 18, 20, 28, 34, 42, 48, 62, 78, 92, 112, 146, 174, 216, 264, 326, 412, 500, 614, 770, 944, 1166, 1444, 1784, 2214, 2730, 3366, 4182, 5164, 6386, 7898, 9770, 12098, 14950, 18488, 22894, 28312, 35020, 43330, 53606

Offset: 1

Author

Gus Wiseman, May 11 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(3) = 2 through a(11) = 8 compositions (empty columns not shown):
  (12)  (23)  (1212)  (34)  (1232)  (45)      (2323)  (56)
  (21)  (32)  (2121)  (43)  (2123)  (54)      (3232)  (65)
                            (2321)  (121212)          (121232)
                            (3212)  (212121)          (123212)
                                                      (212123)
                                                      (212321)
                                                      (232121)
                                                      (321212)
		

Links

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

Crossrefs

Cf. A000079, A008965, A034297, A173258, A325553, A325558, A325590, A325591.

Programs

Mathematica

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

PARI

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(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, R=step(R,n,1); t+=R[n,k]); t)} \\ Andrew Howroyd, Aug 23 2019

Extensions

Terms a(26) and beyond from Andrew Howroyd, Aug 23 2019