A386479 a(0) = 0; thereafter a(n) = 2*n^2 - 3*n + 5.

0, 4, 7, 14, 25, 40, 59, 82, 109, 140, 175, 214, 257, 304, 355, 410, 469, 532, 599, 670, 745, 824, 907, 994, 1085, 1180, 1279, 1382, 1489, 1600, 1715, 1834, 1957, 2084, 2215, 2350, 2489, 2632, 2779, 2930, 3085, 3244, 3407, 3574, 3745, 3920, 4099, 4282, 4469, 4660, 4855, 5054, 5257, 5464, 5675, 5890, 6109, 6332, 6559, 6790

Offset

0,2

Comments

For n>0, a(n) is the maximum number of regions the plane can be divided into by drawing two n-chains (both finite and infinite regions are counted). See A386478 for further information.

We do not at present have an explicit construction that will achieve a(n) for n > 5.

Links

Table of n, a(n) for n=0..59.

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.

N. J. A. Sloane, Two 1-chains (i.e., lines) can divide the plane into at most a(1) = 4 regions; two 2-chains (i.e., V's) can divide the plane into at most a(2) = 7 regions.

N. J. A. Sloane, Illustration for a(3) = 14. Two 3-chains can divide the plane into at most 14 regions.

N. J. A. Sloane, Two 4-chains can divide the plane into at most a(4) = 25 regions. (The two 4-chains are colored respectively black and green.)

N. J. A. Sloane, Two 5-chains can divide the plane into at most a(5) = 40 regions. (The two 5-chains are colored respectively black and red.) It would be nice to have a clearer picture. Region 36 is tiny. Also some of the points where arms of the 5-chains meet are just outside the region shown.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

From Stefano Spezia, Jul 26 2025: (Start)

G.f.: -x*(4-5*x+5*x^2) / (x-1)^3.

E.g.f.: exp(x)*(5 - x + 2*x^2) - 5. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {5, 4, 7}, 60] (* or *) a[n_]:=2n^2-3n+5; Array[a, 60, 0] (* James C. McMahon, Jul 26 2025 *)

Crossrefs

A column of the array in A386478.

Essentially the same (up to offset, initial terms, and the addition of a small constant) as several other sequences, including A014105, A014107, A084849, A096376, A236257, ....

Sequence in context: A221107 A128610 A305124 * A094968 A049946 A076975

Adjacent sequences: A386476 A386477 A386478 * A386480 A386481 A386482

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 25 2025

Extensions

Changed a(0) so as to match changes to A386478. - N. J. A. Sloane, Jul 26 2025

Status

approved