A274112 -id:A274112 - cp's OEIS frontend

A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.

6, 8, 2, 3, 2, 7, 8, 0, 3, 8, 2, 8, 0, 1, 9, 3, 2, 7, 3, 6, 9, 4, 8, 3, 7, 3, 9, 7, 1, 1, 0, 4, 8, 2, 5, 6, 8, 9, 1, 1, 8, 8, 5, 8, 1, 8, 9, 7, 9, 9, 8, 5, 7, 7, 8, 0, 3, 7, 2, 8, 6, 0, 6, 6, 3, 9, 8, 9, 6, 6, 7, 8, 6, 8, 6, 9, 9, 8, 0, 2, 1, 0, 8, 1, 7, 3, 2, 0, 4, 3, 7, 8, 6, 2, 0, 5, 1, 2, 8, 2, 9, 5, 5, 9, 3, 3, 1, 8, 7, 6

Offset: 0

Paul D. Hanna, Oct 24 2015

Constant from Narayana's cows sequence: Limit A000930(n)/A000930(n+1) = r.

Reciprocal of constant described by A092526.

			0.682327803828019327369483739711048256891188581897998577803728606639896...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017. See Corollary 4.22. on p. 24.
Index entries for algebraic numbers, degree 3

Cf. A000930, A025157, A076725, A092526, A274112.

RealDigits[ ((Sqrt[93] + 9)/18)^(1/3) - ((Sqrt[93] - 9)/18)^(1/3), 10, 100][[1]] (* G. C. Greubel, May 01 2017 *)

a(n) = my(r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3)); floor(r*10^(n+1))%10
for(n=0,120,print1(a(n),", "))

solve(r=0, 1,  r^3 + r - 1 ) \\ Michel Marcus, Oct 25 2015

r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3).
Constant r satisfies:
(1) 1/(1 - r*i) = (r + r^2*i) where i^2 = -1.
(2) r = real( 1/(1 - r*i) ).
(3) r = norm( 1/(1 - r*i) ).
(4) r = r^2 + r^4.
Equals 1/A092526. - Vaclav Kotesovec, Nov 27 2017

A274115 Number of equivalence classes of Dyck paths of semilength n for the string duu.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 35, 75, 157, 337, 712, 1529, 3248, 6976, 14869, 31937, 68222, 146536, 313487, 673351, 1441999, 3097326, 6637879, 14257734, 30572032, 65666593, 140860379, 302557585, 649202036, 1394434685, 2992721902, 6428118868, 13798302512, 29637567305, 63626933527, 136665012979

Offset: 0

N. J. A. Sloane, Jun 17 2016

a(n+1) is also the number of Dyck meanders of length n, where catastrophes are allowed. A catastrophe is a direct jump from any altitude > 0 to 0, see the Banderier-Wallner article. - Cyril Banderier, May 30 2019

Gheorghe Coserea, Table of n, a(n) for n = 0..300
Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.
Jean-Luc Baril and Sergey Kirgizov, Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths, arXiv:2104.01186 [math.CO], 2021.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015.

Cf. A000108, A274110, A274111, A274112, A274113, A274114.

A[x_] = 1 + x/(1 + ((1 + x)(Sqrt[1 - 4x^2] - 1))/(2x)) + O[x]^40;
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 27 2018, after Gheorghe Coserea *)

a(n):=if n<2 then 1 else sum((k+1)*sum((binomial(k+1,2*k+2*i-n+3)*binomial(k+2*i,i))/(k+i+1),i,0,(n-k)/2),k,0,n); /* Vladimir Kruchinin, Feb 14 2019 */

seq(N) = {
  my(x='x+O('x^N),
     A000108 = 1+x*Ser(vector(N\2, n, binomial(2*n,n)/(n+1)),'x));
  Vec(1+x/(1 - x*(1+x)*subst(A000108,'x,'x^2)));
};
seq(37)  \\ Gheorghe Coserea, Jan 06 2017

A(x) = 1 + x/(1 - x*(1+x)*A000108(x^2)). - Gheorghe Coserea, Jan 06 2017
a(n) = Sum_{k=0..n} (k+1)*Sum_{i=0..floor((n-k)/2)} C(k+1,2*k+2*i-n+3)*C(k+2*i,i)/(k+i+1), n>1, a(0)=1,a(1)=1. - Vladimir Kruchinin, Feb 14 2019
D-finite with recurrence (-n+1)*a(n) +2*a(n-1) +7*(n-3)*a(n-2) +3*(n-5)*a(n-3) +(-11*n+53)*a(n-4) +4*(-3*n+16)*a(n-5) +4*(-n+6)*a(n-6)=0. - R. J. Mathar, Sep 27 2020

a(0)=1 prepended and more terms from Gheorghe Coserea, Jan 06 2017

A274113 Number of equivalence classes of ballot paths of length n for the string dud.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 11, 16, 26, 39, 63, 95, 154, 235, 381, 585, 948, 1464, 2373, 3682, 5967, 9293, 15060, 23531, 38131, 59741, 96801, 152020, 246310, 387611, 627985, 990027, 1603893, 2532609, 4102726, 6487600, 10509114, 16639214, 26952186, 42722941, 69199472

Offset: 0

N. J. A. Sloane, Jun 17 2016

Gheorghe Coserea, Table of n, a(n) for n = 0..300
K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015, proposition 3.6.

Cf. A274110, A274111, A274112, A274114, A274115.

terms = 43; y[] = 0; Do[y[x] = (-1+x^2-x^4+(2x-3x^2-2x^3+2x^4+2x^5-2x^6) y[x]^2 + (x^2-x^3-x^4) y[x]^3)/(-1+3x+x^2-3x^3-x^4+ x^5) + O[x]^terms, terms]; CoefficientList[y[x], x] (* Jean-François Alcover, Oct 07 2018 *)

x='x; y='y;
Fxy = x^2*(1-x-x^2)*y^3 + 2*x*(1-3/2*x-x^2+x^3+x^4-x^5)*y^2 + (1-3*x-x^2+3*x^3+x^4-3*x^5)*y - 1+x^2-x^4;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(43)  \\ Gheorghe Coserea, Jan 05 2017

G.f. y satisfies: 0 = x^2*(1-x-x^2)*y^3 + 2*x*(1-3/2*x-x^2+x^3+x^4-x^5)*y^2 + (1-3*x-x^2+3*x^3+x^4-3*x^5)*y - 1+x^2-x^4. - Gheorghe Coserea, Jan 05 2017

a(0)=1 prepended by Gheorghe Coserea, Jan 05 2017

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A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.

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A274115 Number of equivalence classes of Dyck paths of semilength n for the string duu.

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A274113 Number of equivalence classes of ballot paths of length n for the string dud.

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