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A325589
Number of compositions of n whose circular differences are all 1 or -1.
6
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
1,3
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
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], SameQ[1, ##]&@@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])) )}
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 11 2019
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, Aug 23 2019
STATUS
approved