Jair Taylor - cp's OEIS frontend

User: Jair Taylor

Jair Taylor has authored 3 sequences.

A212322 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.

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

Offset: 0

Jair Taylor, May 13 2012

nonn

Also known as cyclic Carlitz compositions.

			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.

Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010, pages 87-88.

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.

Removing restriction on the first and last parts gives the Carlitz compositions, A003242.
Row sums of A293595.
Cf. A106369, A241902.

# 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(%);

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 *)

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

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))

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

A182103 Integral of exp(-x)*Phi_n(x) from 0 to infinity, Phi_n the n-th cyclotomic polynomial.

Original entry on oeis.org

0, 2, 4, 3, 34, 2, 874, 25, 727, 20, 4037914, 23, 522956314, 620, 35382, 40321, 22324392524314, 715, 6780385526348314, 39623, 439408062, 3301820, 1177652997443428940314, 40297, 2432903315854636921, 442386620, 6402373706090881, 475412423

Offset: 1

Jair Taylor, Apr 11 2012

nonn

Appears to be all positive integers for n>1, suggesting the possibility of a combinatorial interpretation.

Jair Taylor, Table of n, a(n) for n = 1..497

Agrees with A003422(p) for p prime.

Table[Integrate[Cyclotomic[n, x]/Exp[x], {x, 0, Infinity}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 01 2020 *)

for n in range(1, 40):
  print(integral(e^(-x)*cyclotomic_polynomial(n),x,0,infinity))

A206268 Number of compositions of n with at most one 1.

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 13, 23, 39, 67, 114, 194, 329, 557, 941, 1587, 2672, 4492, 7541, 12643, 21171, 35411, 59166, 98758, 164689, 274393, 456793, 759843, 1263004, 2097872, 3482269, 5776559, 9576639, 15867427, 26276106, 43489802, 71944217, 118958597, 196605701

Offset: 0

Jair Taylor, Feb 18 2012

nonn

			We have a(3) = 3 since 3 = 1 + 2 = 2+1.  A(2) = 1 since 2 is the only composition of 2 that does not have more than one 1.

Jair Taylor, Table of n, a(n) for n = 0..499
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.

CoefficientList[Series[(2 x^3 - 2 x^2 - x + 1)/(x^4 + 2 x^3 - x^2 - 2 x + 1), {x, 0, 38}], x] (* Michael De Vlieger, Dec 09 2020 *)

R. = PowerSeriesRing(QQ)
f = (2*x^3 - 2*x^2 - x + 1)/(x^4 + 2*x^3 - x^2 - 2*x + 1)
print(f.list())

G.f.: (2*x^3 - 2*x^2 - x + 1)/(x^4 + 2*x^3 - x^2 - 2*x + 1).

cp's OEIS Frontend

User: Jair Taylor

A212322 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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A182103 Integral of exp(-x)*Phi_n(x) from 0 to infinity, Phi_n the n-th cyclotomic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

A206268 Number of compositions of n with at most one 1.

Views

Author

Keywords

Examples

Links

Programs

Mathematica

Sage

Formula