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

0,4

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.

Sequences related to inscribed squares

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

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

Sequence in context: A280058 A173548 A006564 * A292022 A265040 A059162

Adjacent sequences: A239349 A239350 A239351 * A239353 A239354 A239355

Keyword

nonn,easy

Author

Jonathan Sondow, Mar 21 2014

Status

approved