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%I #7 Aug 23 2019 13:36:01
%S 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,
%T 146,174,216,264,326,412,500,614,770,944,1166,1444,1784,2214,2730,
%U 3366,4182,5164,6386,7898,9770,12098,14950,18488,22894,28312,35020,43330,53606
%N Number of compositions of n whose circular differences are all 1 or -1.
%C A composition of n is a finite sequence of positive integers summing to n.
%C 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).
%H Andrew Howroyd, <a href="/A325589/b325589.txt">Table of n, a(n) for n = 1..200</a>
%e The a(3) = 2 through a(11) = 8 compositions (empty columns not shown):
%e (12) (23) (1212) (34) (1232) (45) (2323) (56)
%e (21) (32) (2121) (43) (2123) (54) (3232) (65)
%e (2321) (121212) (121232)
%e (3212) (212121) (123212)
%e (212123)
%e (212321)
%e (232121)
%e (321212)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[Differences[Append[#,First[#]]]]&]],{n,15}]
%o (PARI)
%o 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])) )}
%o 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
%Y Cf. A000079, A008965, A034297, A173258, A325553, A325558, A325590, A325591.
%K nonn
%O 1,3
%A _Gus Wiseman_, May 11 2019
%E Terms a(26) and beyond from _Andrew Howroyd_, Aug 23 2019
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