Kobon Triangle

Author

Eric W. Weisstein
July 27, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/KobonTriangle.nb.

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

Sources

https://oeis.org/A006066

https://web.archive.org/web/20120503024433/http://www.maa.org/editorial/mathgames/mathgames_02_08_06.html

https://oeis.org/A006066/a006066_2.pdf

Baltholdi et al. 2007.  https://arxiv.org/abs/0706.0723

Clément and Bader.  "Tighter Upper Bound for the Number of Kobon Triangles."  https://oeis.org/A006066/a006066.pdf

Savchuk.  https://arxiv.org/pdf/2507.07951

Exact values

Baltholdi et al. 2007.  https://arxiv.org/abs/0706.0723

TextGrid[Table[{n=2 (2^t)+1,n(n-2)/3},{t,0,10}],Dividers->All,Alignment->Decimal]

({{3, 1}, {5, 5}, {9, 21}, {17, 85}, {33, 341}, {65, 1365}, {129, 5461}, {257, 21845}, {513, 87381}, {1025, 349525}, {2049, 1398101}})

TextGrid[Table[{n=14 (2^t)+1,n(n-2)/3},{t,0,10}],Dividers->All,Alignment->Decimal]

TextGrid[Table[{n=6 (2^t)+1,(n(n-2)-2)/3},{t,0,10}],Dividers->All,Alignment->Decimal]

Upper bounds

Tamura

A032765

Tamura[n_]:=Floor[((n(n-2))/3)]

seq1=Table[Floor[((n(n-2))/3)],{n,20}]

FindLinearRecurrence[seq1]

gf=FindGeneratingFunction[Rest@seq1,x]

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

Table[(3 n (n+2)+2 Cos[((2 n Pi)/3)]+2 (Sqrt[3]) Sin[((2 n Pi)/3)]-2)/9,{n,0,20}]

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

ClementBader

Typo; last case should be (n^2-2n-2)/3

Table[Floor[((n(n-2))/3)]-((n^2-2n-2)/3),{n,Select[Range[100],MatchQ[Mod[#,6],1|4]&]}]//Union

Table[Floor[((n(n-2))/3)]-(((((n-1))^2)/3)-1),{n,Select[Range[100],MatchQ[Mod[#,6],1|4]&]}]//Union

Expand[((n^2-2n-2)/3)-(((((n-1))^2)/3)-1)]

ClementBaderCorrected[n_]:=Switch[Mod[n,6],
3|5,n(n-2)/3,
0|2,n(n-2)/3-1,
1|4,(n^2-2n-2)/3] (* corrected *)

seq2=Table[ClementBaderCorrected[n],{n,50}]

coef=FindLinearRecurrence[seq2]

OEISRecurrenceIndexString[coef]

OEISRecurrenceString[coef]

Length[coef]

Take[seq2,8]

LinearRecurrence[{2,-1,0,0,0,1,-2,1},{-1,-1,1,2,5,7,11,15},50]

gf=FindGeneratingFunction[seq2,x]//Factor

OEISGeneratingFunctionString[gf,x]

FullSimplify[SeriesCoefficient[((1-x-2 (x^2)+(x^3)-2 (x^4)+(x^5)-3 (x^6)+(x^7))/((((-1+x))^3) (1+x) (1-x+(x^2)) (1+x+(x^2)))),{x,0,n}],n>=0&&n∈Integers]

FullSimplify[SeriesCoefficient[((1-x-2 (x^2)+(x^3)-2 (x^4)+(x^5)-3 (x^6)+(x^7))/((((-1+x))^3) (1+x) (1-x+(x^2)) (1+x+(x^2))))x,{x,0,n}],n>=0&&n∈Integers]

Table[(1/9) (-8+3 (((-1))^n)+3 (n^2)-ChebyshevT[n,-(1/2)]-3 ChebyshevT[n,(1/2)]),{n,20}]

Baltholdi

BaltholdiOdd[n_]:=If[EvenQ[n],Floor[(n/3)(n-(7/3))],0]

Table[Floor[(n/3)(n-(7/3))],{n,0,50}]

Table

TextGrid[Transpose@Prepend[
Table[{n,Tamura[n],ClementBaderCorrected[n],BaltholdiOdd[n]},{n,3,20}],
Thread[Style[{Style["n",Italic],"Tamura","Clement\[Hyphen]Bader","Baltholdi"},Bold]]],
Dividers->All,Alignment->{{Left,{Decimal}}}]

f[n_?OddQ]:=Floor[((n(n-2))/3)]
f[n_?EvenQ]:=Floor[(n/3)(n-(7/3))]

Table[f[n],{n,15}]

{
Plot[{((n(n-2))/3),(n/3)(n-(7/3))},{n,1,10}],
Plot[Evaluate@Floor[{((n(n-2))/3),(n/3)(n-(7/3))}],{n,1,10}]
}

Table[Floor[((n(n-2))/3)],{n,15}]

Table[Floor[(n/3)(n-(7/3))],{n,15}]

Assuming[n>2,Simplify[((n(n-2))/3)>(n/3)(n-(7/3))]]

Table[Floor[((n(n-2))/3)]-Floor[(n/3)(n-(7/3))],{n,200}]

Figures

Savchuk

Import["https://arxiv.org/pdf/2507.07951",{"PageGraphics",9}]

Import["https://arxiv.org/pdf/2507.07951",{"PageGraphics",{12,13,14}}]

10

Import["https://oeis.org/A006066/a006066.tiff"]

Serhiy Grabarchuk

Import@"https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon10-25.gif"

Viatcheslav Kabanovitch

Import["https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon10-25-2.gif"]

11

Viatcheslav Kabanovitch

Import@"https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon11-32.gif"

12

Viatcheslav Kabanovitch

Import["https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon12-38.gif"]

11

Import@"https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon10-25.gif"

12

Import@"https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon10-25-2.gif"

13

Import@"https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/Kobon13-47.gif"

15

Viatcheslav Kabanovitch

Import["https://web.archive.org/web/20120503024433im_/http://www.maa.org/editorial/mathgames/ToshitakaSuzuki.gif"]

Import@"https://web.archive.org/web/20120503024433/http://www.maa.org/editorial/mathgames/ToshitakaSuzuki.svg"

17

Clément and Bader (2007)

Cl\'ement, G.\ and Bader, J.  ``Tighter Upper Bound for the Number of Kobon Triangles.''  Dec.~21, 2007.  https://oeis.org/A006066/a006066.pdf

Import["https://oeis.org/A006066/a006066.pdf","Elements"]

Import["https://oeis.org/A006066/a006066.pdf","EmbeddedImages"]

Import["https://oeis.org/A006066/a006066.pdf",{"PageGraphics",1}]