A325557 Number of compositions of n with equal differences up to sign.

1, 1, 2, 4, 6, 8, 13, 12, 20, 24, 25, 29, 49, 40, 50, 64, 86, 80, 105, 102, 164, 175, 186, 208, 325, 316, 382, 476, 624, 660, 814, 961, 1331, 1500, 1739, 2140, 2877, 3274, 3939, 4901, 6345, 7448, 9054, 11157, 14315, 17181, 20769, 25843, 32947, 39639, 48257, 60075

Offset

0,3

Comments

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

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

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

Example

The a(1) = 1 through a(8) = 20 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)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (123)     (61)       (62)
                            (11111)  (141)     (151)      (71)
                                     (222)     (232)      (161)
                                     (321)     (313)      (242)
                                     (1212)    (12121)    (323)
                                     (2121)    (1111111)  (1232)
                                     (111111)             (1313)
                                                          (2123)
                                                          (2222)
                                                          (2321)
                                                          (3131)
                                                          (3212)
                                                          (21212)
                                                          (11111111)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], SameQ@@Abs[Differences[#]]&]], {n, 0, 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, s)={my(R=matid(n), t=0); while(R, R=step(R, n, s); t+=vecsum(R[n, ])); t}
a(n) = {numdiv(max(1, n)) + sum(s=1, n-1, w(n, s))} \\ Andrew Howroyd, Aug 22 2019

Crossrefs

Cf. A000079, A047966, A049988, A070211, A098504, A173258, A175342, A325545, A325546, A325547, A325548, A325552, A325558.

Sequence in context: A187779 A086141 A067883 * A283769 A338352 A117117

Adjacent sequences: A325554 A325555 A325556 * A325558 A325559 A325560

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