A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

0, 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

0,3

Comments

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).

Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.

Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).

The partial sum of this sequence is A184985.

a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.

a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.

Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

References

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.

V. Manturov, Knot Theory, CRC Press, 2004.

Links

Table of n, a(n) for n=0..85.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.

J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.

A. Banerjee, Knot theory [Foil knot family].

D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.

L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.

Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

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

Formula

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.

a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).

a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).

a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.

A005803(a(n)) = A005096(a(n)) = A000007(n).

A107583(a(n)) = A103775(n+5).

a(n+1) = 2^A185012(n+1), with a(0) = 0.

a(n) = A163985(n) mod A004278(n+1).

a(n) = A157928(n) + A171386(n+1).

a(n) = A063524(n) + A157928(n) + A185012(n).

a(n) = A010701(n) - A141044(n) - A179184(n).

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

E.g.f.: (2*exp(x) - 2 + x^2)/2.

Example

For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

Mathematica

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0, 1, 2}, 120, {1}] (* Harvey P. Dale, Feb 20 2023 *)

Prog

(PARI) a(n) = if(n>2, 1, n);
(Maxima) makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

Crossrefs

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

Sequence in context: A299912 A369162 A329684 * A054977 A272901 A078315

Adjacent sequences: A294616 A294617 A294618 * A294620 A294621 A294622

Keyword

nonn,easy

Author

Franck Maminirina Ramaharo, Nov 05 2017

Status

approved