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%I #49 Mar 07 2020 08:49:55
%S 1,1,1,3,3,5,13,17,29,55,99,161,293,507,881,1561,2727,4743,8337,14579,
%T 25497,44675,78173,136753,239437,419077,733377,1283701,2246823,
%U 3932249,6882603,12046313,21083545,36901587,64586887,113042011,197851265,346287829,606086169
%N Number of compositions of n such that no two adjacent parts are equal, and the first part is not equal to the last part if there is more than one part.
%C Also known as cyclic Carlitz compositions.
%D Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010, pages 87-88.
%H Alois P. Heinz, <a href="/A212322/b212322.txt">Table of n, a(n) for n = 0..1000</a> (first 200 terms from Jair Taylor)
%H P. Hadjicostas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Hadjicostas/hadji5.html">Cyclic, dihedral and symmetric Carlitz compositions of a positive integer</a>, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
%H Jair Taylor, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p1">Counting Words with Laguerre Series</a>, Electron. J. Combin., 21 (2014), P2.1.
%F G.f.: 1 + sum(k>0: x^k/(1+x^k)^2)/(1-sum(k>0, x^k/(1+x^k))) + sum(k>0, x^(2k)/(1+x^k)).
%F a(n) ~ c * d^n, where d = 1.750241291718309031249738624639... (see A241902), c = 0.350601274598240344779505805689.... - _Vaclav Kotesovec_, May 01 2014
%e The cyclic Carlitz compositions of the n = 1...6 are
%e 1;
%e 2;
%e 12, 21, 3;
%e 13, 31, 4;
%e 14, 23, 32, 41,5;
%e 1212, 123, 132, 15, 2121, 213, 231, 24, 312, 321, 42, 51, 6.
%p # For getting the first M-1 terms, from _N. J. A. Sloane_, Apr 26 2014
%p M:=101:
%p t1:=add(x^i/(1+x^i),i=1..M):
%p t2:=add(x^i/(1+x^i)^2,i=1..M):
%p t3:=add(x^(2*i)/(1+x^i),i=1..M):
%p t0:=t2/(1-t1)+t3:
%p series(t0,x,30);
%p seriestolist(%);
%t terms = 39;
%t gf = 1 + Sum[x^k/(1 + x^k)^2, {k, 1, terms}]/(1 - Sum[x^k/(1 + x^k), {k, 1, terms}]) + Sum[x^(2 k)/(1 + x^k), {k, 1, terms}] + O[x]^terms;
%t CoefficientList[gf, x] (* _Jean-François Alcover_, Dec 30 2017 *)
%o (Sage)
%o for n in range(15):
%o Q = []
%o for comp in Compositions(n) :
%o if len(comp) == 1 or all(comp[k] != comp[k+1] for k in range(-1,len(comp)-1)):
%o Q.append(comp)
%o print(len(Q))
%o (PARI)
%o a(n) = { polcoeff(1 + sum(k=1, n, x^k/(1+x^k)^2 + O(x*x^n))/(1-sum(k=1, n, x^k/(1+x^k) + O(x*x^n))) + sum(k=1, n, x^(2*k)/(1+x^k) + O(x*x^n)), n) } \\ _Andrew Howroyd_, Oct 14 2017
%Y Removing restriction on the first and last parts gives the Carlitz compositions, A003242.
%Y Row sums of A293595.
%Y Cf. A106369, A241902.
%K nonn
%O 0,4
%A _Jair Taylor_, May 13 2012
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