OFFSET

0,4

COMMENTS

In 1911 Toeplitz conjectured the Square Peg (or Inscribed Square) Problem: Every continuous simple closed curve in the plane contains 4 points that are the vertices of a square. The conjecture is still open. Many special cases have been proved; see Matschke's beautiful 2014 survey.

Recently van Heijst proved that any real algebraic curve in R^2 of degree d inscribes either at most (d^4 - 5d^2 + 4d)/4 or infinitely many squares. He conjectured that a generic complex algebraic plane curve inscribes exactly (d^4 - 5d^2 + 4d)/4 squares.

REFERENCES

Otto Toeplitz, Über einige Aufgaben der Analysis situs, Verhandlungen der Schweizerischen Naturforschenden Gesellschaft in Solothurn, 4 (1911), 197.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

Wouter van Heijst, The algebraic square peg problem, arXiv:1403.5979 [math.AG], 2014.

Wouter van Heijst, The algebraic square peg problem, Master’s thesis, Aalto University, 2014.

Benjamin Matschke, A Survey on the Square Peg Problem, AMS Notices, 61 (2014), 346-352.

Benjamin Matschke, Extended Survey on the Square Peg Problem, Max Planck Institute for Mathematics, 2014.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = (n^4 - 5*n^2 + 4*n)/4 = n*(n - 1)*(n^2 + n - 4)/4 = A000217(n-1)*A034856(n-1), which shows the formula is an integer.

G.f.: x^2 * (1 + 7*x - 2*x^2) / (1 - x)^5. - Michael Somos, Mar 21 2014

a(n) = A172225(n)/2. - R. J. Mathar, Jan 09 2018

EXAMPLE

A point or a line has no inscribed squares, so a(0) = a(1) = 0.

A circle has infinitely many inscribed squares, and an ellipse that is not a circle has exactly one, agreeing with a(2) = 1.

G.f. = x^2 + 12*x^3 + 48*x^4 + 130*x^5 + 285*x^6 + 546*x^7 + 952*x^8 + ...

MATHEMATICA

Table[(n^4 - 5 n^2 + 4 n)/4, {n, 0, 38}]

PROG

(PARI) for(n=0, 50, print1((n^4 - 5*n^2 + 4*n)/4, ", ")) \\ G. C. Greubel, Aug 07 2018

(Magma) [(n^4 - 5*n^2 + 4*n)/4: n in [0..50]]; // G. C. Greubel, Aug 07 2018

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Jonathan Sondow, Mar 21 2014

STATUS

approved