A273724 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 3i != i mod n, draw an (undirected) chord from i to (3i mod n). Then a(n) is the total number of distinct chords.

0, 0, 0, 2, 1, 4, 4, 6, 3, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 19, 24, 24, 26, 25, 28, 28, 30, 27, 32, 32, 34, 33, 36, 36, 38, 35, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 51, 56, 56, 58, 57, 60, 60, 62, 59, 64, 64, 66, 65, 68, 68, 70, 67, 72, 72, 74, 73, 76, 76, 78, 75, 80, 80

Offset

0,4

Links

Brooke Logan and N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Kival Ngaokrajang, Illustration of initial terms

Burkard Polster, Times Tables, Mandelbrot and the Heart of Mathematics, Mathologer video (2015).

Formula

a(n) = n-1 if n>0 is odd, n-2 if n == +-2 (mod 8), n-3 if n == 4 (mod 8), and n-5 if n == 0 (mod 8). These formulas are easily established by observing that the chord at i is missing if 2i == 0 mod n, and the chords starting at i and at 3i coincide if 8i == 0 mod n. The formulas then imply that the g.f. is 4+x^2/(1-x)^2-(4+x^2+2*x^4+x^6)/(1-x^8), which can be rewritten as (5*x^63*x^5+2*x^4+3*x^2-x+2)*x^3/((1-x)*(1-x^8)). (This g.f. was conjectured by Colin Barker.) - Brooke Logan and N. J. A. Sloane, Jun 23 2016

a(n) = a(n-1)+a(n-8)-a(n-9) for n>9. - Colin Barker, May 29 2016 (This follows from the above g.f. - Brooke Logan and N. J. A. Sloane)

Prog

(Small Basic)
For n=0 to 200
For i = 0 To n-1
  x[i]=0
  a=i*3
  b=a-n
  c=a-2*n
  If b<0 And c<0 Then
    x[i]=a
  EndIf
  If b>0 And c<0 Then
    x[i]=b
  EndIf
  If b>0 And c>0 Then
    x[i]=c
  EndIf
EndFor
c=0
For j = 0 To n-1
  For k=0 To j
    If x[j]=k And x[k]=j Then
      c=c+1
    EndIf
  EndFor
EndFor
aa=n-c
TextWindow.Write(aa+", ")
Endfor

Crossrefs

Cf. A117571 (if 3i is changed to 2i), A274462 (if 3i is changed to 4i).

Sequence in context: A326147 A351168 A196082 * A108755 A093049 A326146

Adjacent sequences: A273721 A273722 A273723 * A273725 A273726 A273727

Keyword

nonn

Author

Kival Ngaokrajang, May 28 2016

Extensions

Definition edited by N. J. A. Sloane, Jun 23 2016

Status

approved