A389624 Maximum number of regions that can be formed in the plane by drawing n lollipop (or qoppa) shapes of any size and orientation.

1, 2, 10, 25, 45, 71, 104, 142, 186, 237, 294, 356, 425, 500, 580, 667, 761, 859

Offset

0,2

Comments

A lollipop or qoppa shape is a circle with a half-line attached to a point of the circle and extending from there in a radial direction to infinity.

The values a(0), ..., a(4) are due to Cutler-Karlsson-Sloane, the values a(5), a(6), a(7), a(10) are due to Paulsen, and the values a(8), a(9), a(11), ..., a(17) are due to Mahajan-Chopra.

Moreover, Mahajan-Chopra show that a(18) is 964 or 965, that a(19) = 1076, and that a(20) is 1193 or 1194. - Matthias Paulsen, Jun 05 2026

Links

Table of n, a(n) for n=0..17.

The Chordettes, "Lollipop, lollipop, oh lolli, lolli, lolli, lollipop...", Youtube video, 2006.

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, Figure 39: Illustrating a(2) = 10

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, Figure 40: Illustrating a(3) = 25

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, Figure 41: Illustrating a(4) = 45. The figure has 40 crossings and divides the plane into 45 regions.

Jonas Karlsson, Illustrating a(4) = 45 (i). The figure has 40 crossings and divides the plane into 45 regions. In this view only three of the four lollipops are visible to the naked eye. The fourth lollipop is located where the three circles come together, as can be seen in the next link.

Jonas Karlsson, Illustrating a(4) = 45 (ii). Closeup of the region where the three large circles come together. In this view the three large circles are so enormous that the intersections of the circles and their stems look like "T"s (the arcs of the circles appear to be straight lines).

Jonas Karlsson, Illustrating a(4) = 45 (iii). Python/mpmath code giving coordinates for the four lollipop arrangement.

Siddhartha Mahajan and Paras Chopra, A Two-Graph Refinement of Paulsen's Lollipop Bounds, arXiv:2606.06064[math.CO], 2026.

Matthias Paulsen, The Lollipop Problem.

Siobhan Roberts, Infinite Pancakes, Anyone? A team of mathematicians seeks to cut an infinitely large pancake into as many pieces as possible, in a new take on an old puzzle., New York Times, Jan 20 2026.

N. J. A. Sloane, The Lollipop Problem, Dec. 25-26, 2025. [Superseded by the Cutler-Karlsson-Sloane and Paulsen papers.]

Crossrefs

Cf. A000124, A386480, A386486, A389608.

See A396714 for the equal-radii case.

Sequence in context: A336958 A305600 A396714 * A389608 A058373 A167386

Adjacent sequences: A389621 A389622 A389623 * A389625 A389626 A389627

Keyword

nonn,more

Author

N. J. A. Sloane, Dec 26 2025

Extensions

Thanks to Harry E. Neel and another member of the SeqFan mailing list for the Chordettes link.

Updated by N. J. A. Sloane, Jan 30 2026 and Jun 03 2026

a(8)-a(17) from the Mahajan-Chopra paper added by Matthias Paulsen, Jun 05 2026

Status

approved