Rectilinear Local Crossing Number

Author

Eric W. Weisstein
July 1, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/RectilinearLocalCrossingNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RectilinearLocalCrossingNumber.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Classes

Complete

A374243

https://doi.org/10.1016/j.jcta.2017.04.003

Table[Ceiling[(1/2)(n-3-Ceiling[((n-3)/3)])Ceiling[((n-3)/3)]],{n,20}]

lcr[n_]:=Ceiling[(1/2)(n-3-Ceiling[((n-3)/3)])Ceiling[((n-3)/3)]]

TextGrid[Table[{n,lcr[n],Piecewise[{{4,n==8},{15,n==14}}]},{n,{8,14}}],Dividers->All,Alignment->Decimal]

seq=Table[lcr[n],{n,60}]

ToString[seq]//StringLength

FindSequenceFunction[seq,n]//Head//Timing

gf=x FindGeneratingFunction[seq,x,ValidationLength->2]

StringReplace[ToString[gf,InputForm]," "->""]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0&&n∈Integers]

Table[(1/108) (89-9 (((-1))^n)+6 n (-11+2 n)+9 Cos[((n π)/3)]+(19-6 n) Cos[((2 n π)/3)]-9 (Sqrt[3]) Sin[((n π)/3)]+(Sqrt[3]) (-21+2 n) Sin[((2 n π)/3)]),{n,60}]==seq

CoefficientList[Series[(((x^4) (-1-(x^2)-(x^4)-(x^5)))/((((-1+x))^3) (((1+x+(x^2)))^2) (1+(x^3)))),{x,0,20}],x]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length@coef]

LinearRecurrence[{1,0,1,-1,0,1,-1,0,-1,1},{0,0,0,0,1,1,2,3,4,6},20]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

FullSimplify[RSolve[Prepend[Table[p[k,x]==seq[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n>0&&n∈Integers]