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A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

%I A325591 #12 Sep 21 2019 02:52:00
%S A325591 1,2,4,6,11,15,27,43,68,116,189,311,519,860,1433,2380,3968,6613,11018,
%T A325591 18374,30633,51089,85208,142113,237055,395409,659576,1100262,1835382,
%U A325591 3061711,5107445,8520122,14213135,23710173,39553138,65982316,110071459,183620990,306316328
%N A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.
%C A325591 A composition of n is a finite sequence of positive integers summing to n.
%C A325591 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 A325591 Andrew Howroyd, <a href="/A325591/b325591.txt">Table of n, a(n) for n = 1..200</a>
%F A325591 a(n) ~ c * d^n, where d = 1.66820206701846111636107... (see A034297), c = 0.65837031047271348106444... - _Vaclav Kotesovec_, Sep 21 2019
%e A325591 The a(1) = 1 through a(6) = 15 compositions:
%e A325591   (1)  (2)   (3)    (4)     (5)      (6)
%e A325591        (11)  (12)   (22)    (23)     (33)
%e A325591              (21)   (112)   (32)     (222)
%e A325591              (111)  (121)   (122)    (1122)
%e A325591                     (211)   (212)    (1212)
%e A325591                     (1111)  (221)    (1221)
%e A325591                             (1112)   (2112)
%e A325591                             (1121)   (2121)
%e A325591                             (1211)   (2211)
%e A325591                             (2111)   (11112)
%e A325591                             (11111)  (11121)
%e A325591                                      (11211)
%e A325591                                      (12111)
%e A325591                                      (21111)
%e A325591                                      (111111)
%t A325591 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[DeleteCases[Differences[Append[#,First[#]]],0]]&]],{n,15}]
%o A325591 (PARI)
%o A325591 step(R,n,D)={matrix(n, n, i, j, if(i>j, sum(k=1, #D, my(s=D[k]); if(j>s && j+s<=n, R[i-j, j-s]))) )}
%o A325591 a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)<=1), t=0); while(R, t+=R[n,k]; R=step(R,n,[0,1,-1])); t)} \\ _Andrew Howroyd_, Aug 23 2019
%Y A325591 Row sums of A309931.
%Y A325591 Cf. A000079, A008965, A034297, A173258, A325351, A325551, A325553, A325558, A325589, A325590.
%K A325591 nonn
%O A325591 1,2
%A A325591 _Gus Wiseman_, May 12 2019
%E A325591 Terms a(21) and beyond from _Andrew Howroyd_, Aug 23 2019