Losanitsch's Triangle

Author

Eric W. Weisstein
July 11, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/LosanitschsTriangle.nb.

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

a[n_,0]:=1
a[n_,n_]:=1
a[n_,k_]:=a[n,k]=a[n-1,k-1]+a[n-1,k]-Binomial[n/2-1,(k-1)/2]Mod[k,2]Mod[n-1,2]

ColumnForm[Table[a[n,k],{n,0,8},{k,0,n}],Center]

Closed Form

RSolve[{
a[n,0]==1,
a[n,n]==1,a[n,k]==a[n-1,k-1]+a[n-1,k]-Binomial[n/2-1,(k-1)/2]Mod[k,2]Mod[n-1,2]},a[n,k],{n,k}]

b[n_,k_]:=(1/2)(Binomial[n,k]+Binomial[Mod[n,2],Mod[k,2]]Binomial[Floor[n/2],Floor[k/2]])

ColumnForm[Table[b[n,k],{n,0,8},{k,0,n}],Center]

Sums

Row Sums

Table[(2^(n-2))+(2^(Floor[n/2]-1)),{n,10}]

Diagonal Sums

(Subscript[D, 2m])

Table[Sum[b[n,m-n],{n,0,m}],{m,0,20,2}]

Table[(1/2)(Fibonacci[2m-1]+Fibonacci[m+1]),{m,10}]

Sum[b[n,m-n],{n,0,m}]

(Subscript[D, 2m-1])

Table[Sum[b[n,m-n],{n,0,m}],{m,1,20,2}]

Table[(1/2)(Fibonacci[2m]+Fibonacci[m]),{m,10}]

Sum[b[n,m-n],{n,0,m}]

Binary Plot

<<Utilities`Plot`

Length[t=Flatten[Table[Table[a[n,k],{n,0,35},{k,0,n}]]]]

Show[PixelPerfect[BasePlot[Take[t,255]],BlockSize->2]]

Show[PixelPerfect[BasePlot[Take[t,511]],BlockSize->1]]