A325558 Number of compositions of n with equal circular differences up to sign.

1, 2, 4, 5, 6, 10, 8, 16, 13, 16, 18, 32, 20, 30, 30, 57, 34, 52, 46, 96, 74, 86, 84, 174, 119, 170, 192, 306, 244, 332, 372, 628, 560, 694, 812, 1259, 1228, 1566, 1852, 2696, 2806, 3538, 4260, 5894, 6482, 8098, 9890, 13392, 15049, 18706, 23018, 30298, 35198

Offset

1,2

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).

Links

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

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (222)     (61)       (62)
                                     (1212)    (1111111)  (71)
                                     (2121)               (1232)
                                     (111111)             (1313)
                                                          (2123)
                                                          (2222)
                                                          (2321)
                                                          (3131)
                                                          (3212)
                                                          (11111111)

Mathematica

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

Prog

(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])) )}
w(n, k, s)={my(R=matrix(n, n, i, j, i==j&&abs(i-k)==s), t=0); while(R, R=step(R, n, s); t+=R[n, k]); t}
a(n) = {numdiv(max(1, n)) + sum(s=1, n-1, sum(k=1, n, w(n, k, s)))} \\ Andrew Howroyd, Aug 22 2019

Crossrefs

Cf. A000079, A008965, A047966, A049988, A098504, A173258, A175342, A325553, A325557, A325588, A325589.

Sequence in context: A014224 A175342 A077312 * A140779 A348937 A117890

Adjacent sequences: A325555 A325556 A325557 * A325559 A325560 A325561

Keyword

nonn

Author

Gus Wiseman, May 11 2019

Extensions

a(26)-a(42) from Lars Blomberg, May 30 2019

Terms a(43) and beyond from Andrew Howroyd, Aug 22 2019

Status

approved