A294619 -id:A294619 - cp's OEIS frontend

A329684 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and HH.

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Valerie Roitner, Nov 29 2019

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
This sequence is periodic with a pre-period of length 3 (namely 1, 1, 2) and a period of length 1 (namely 1).
Decimal expansion of 1009/9000. - Elmo R. Oliveira, Jun 16 2024

			a(2)=2 since UD and HH are allowed. For n different from 2, only the excursion H^n is allowed.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A329680, A329682, A329683.

Essentially the same as A294619, A261143 and A141044.

PadRight[{1, 1, 2}, 100, 1] (* Paolo Xausa, Aug 28 2024 *)

G.f.: (1+t^2-t^3)/(1-t).

For n >= 0, a(2) = 2, otherwise a(n) = 1. - Elmo R. Oliveira, Jun 16 2024

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 24 2018

Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves.

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 1, 1, ...
  1, 1, 3, 2, 2, 2, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  ...

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on this sequence and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

Cf. A300453, A300454, A316989, A321126, A321127.

b[n_] = If[n == 0 || n == 2, 1, 0];
T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten

b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
create_list(T(k, n - k), n, 0, 12, k, 0, n);

T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n).
T(n,1) = A300453(n+1,A321126(n,1)).
T(n,2) = A300454(n,A321126(n,2)).
T(n,n) = A321127(n,A004280(n+1)).
G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).
E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.

A373565 Expansion of x + 1/(1 - x).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Jun 10 2024

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A294619.

CoefficientList[Series[x + 1/(1 - x), {x, 0, 105}], x] (* Michael De Vlieger, Jun 10 2024 *)

a(n) = [x^n] (x^2 - x - 1) / (x - 1).

E.g.f.: exp(x) + x. - Stefano Spezia, Jun 10 2024

cp's OEIS Frontend

A329684 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and HH.

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A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

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Programs

Mathematica

Maxima

Formula

A373565 Expansion of x + 1/(1 - x).

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cp's OEIS Frontend

A329684 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and HH.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A373565 Expansion of x + 1/(1 - x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).