A239352 van Heijst's upper bound on the number of squares inscribed by a real algebraic curve in R^2 of degree n, if the number is finite.

0, 0, 1, 12, 48, 130, 285, 546, 952, 1548, 2385, 3520, 5016, 6942, 9373, 12390, 16080, 20536, 25857, 32148, 39520, 48090, 57981, 69322, 82248, 96900, 113425, 131976, 152712, 175798, 201405, 229710, 260896, 295152, 332673, 373660, 418320, 466866, 519517

Offset: 0

Author

Jonathan Sondow, Mar 21 2014

Keywords

• nonn
• easy

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.

Examples

			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 + ...
		

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.
• Sequences related to inscribed squares
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A088544, A089058, A123673, A123697, A209432, A231739.

Programs

Magma

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

Mathematica

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

PARI

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

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