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A212322
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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.
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5
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1, 1, 1, 3, 3, 5, 13, 17, 29, 55, 99, 161, 293, 507, 881, 1561, 2727, 4743, 8337, 14579, 25497, 44675, 78173, 136753, 239437, 419077, 733377, 1283701, 2246823, 3932249, 6882603, 12046313, 21083545, 36901587, 64586887, 113042011, 197851265, 346287829, 606086169
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OFFSET
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0,4
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COMMENTS
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Also known as cyclic Carlitz compositions.
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REFERENCES
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Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010, pages 87-88.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 200 terms from Jair Taylor)
P. Hadjicostas, Cyclic, dihedral and symmetric Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
Jair Taylor, Counting Words with Laguerre Series, Electron. J. Combin., 21 (2014), P2.1.
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FORMULA
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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)).
a(n) ~ c * d^n, where d = 1.750241291718309031249738624639... (see A241902), c = 0.350601274598240344779505805689.... - Vaclav Kotesovec, May 01 2014
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EXAMPLE
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The cyclic Carlitz compositions of the n = 1...6 are
1;
2;
12, 21, 3;
13, 31, 4;
14, 23, 32, 41,5;
1212, 123, 132, 15, 2121, 213, 231, 24, 312, 321, 42, 51, 6.
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MAPLE
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# For getting the first M-1 terms, from N. J. A. Sloane, Apr 26 2014
M:=101:
t1:=add(x^i/(1+x^i), i=1..M):
t2:=add(x^i/(1+x^i)^2, i=1..M):
t3:=add(x^(2*i)/(1+x^i), i=1..M):
t0:=t2/(1-t1)+t3:
series(t0, x, 30);
seriestolist(%);
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MATHEMATICA
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terms = 39;
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;
CoefficientList[gf, x] (* Jean-François Alcover, Dec 30 2017 *)
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PROG
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(Sage)
for n in range(15):
Q = []
for comp in Compositions(n) :
if len(comp) == 1 or all(comp[k] != comp[k+1] for k in range(-1, len(comp)-1)):
Q.append(comp)
print(len(Q))
(PARI)
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
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CROSSREFS
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Removing restriction on the first and last parts gives the Carlitz compositions, A003242.
Row sums of A293595.
Cf. A106369, A241902.
Sequence in context: A320176 A298478 A144419 * A226921 A133190 A052898
Adjacent sequences: A212319 A212320 A212321 * A212323 A212324 A212325
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KEYWORD
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nonn
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AUTHOR
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Jair Taylor, May 13 2012
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STATUS
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approved
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