A389614 a(n) = 2*n^2 + 1 - (m*n - 3*m*(m+1)/2) where m = floor(n/3).

1, 3, 9, 19, 32, 49, 70, 94, 122, 154, 189, 228, 271, 317, 367, 421, 478, 539, 604, 672, 744, 820, 899, 982, 1069, 1159, 1253, 1351, 1452, 1557, 1666, 1778, 1894, 2014, 2137, 2264, 2395, 2529, 2667, 2809, 2954, 3103, 3256, 3412, 3572, 3736, 3903, 4074, 4249, 4427, 4609, 4795, 4984, 5177, 5374, 5574, 5778, 5986, 6197, 6412

Offset

0,2

Comments

From N. J. A. Sloane, Dec 19 2025 (Start)

Let Tbar denote a planar graph consisting of a distinguished point C (the center), an infinite line L through C, and a semi-infinite line through C perpendicular to L. Let M(n) denote the maximum number of regions that the plane can be divided into by drawing n Tbars. Cutler and Sloane (2025) conjectured that M(n) = a(n) for all n. This conjecture has now been established by Jonas Karlsson, see the reference. - N. J. A. Sloane, Jan 29 2026

The links below contain numerical coordinates for the best arrangements we have found for 2 <= n <= 12 Tbars.

In the links, each Tbar is specified by three numbers: the coordinates (x,y) of C, and the angle theta of the semi-infinite line at C, measured in radians from the horizontal. Because of a quirk in the optimizer, the values of theta may exceed 2*pi (they have not been reduced modulo 2*pi).

In the links, for a graph formed from n Tbars, V denotes the total number of points where the Tbars cross each other, and R = V + 2*n + 1 is the number of regions in the graph. Provided no round-off errors have occurred, R should equal a(n).

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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, Coordinates for 2 Tbars (see Comments)

David O. H. Cutler, Coordinates for 3 Tbars (see Comments)

David O. H. Cutler, Coordinates for 4 Tbars (see Comments)

David O. H. Cutler, Coordinates for 5 Tbars (see Comments)

David O. H. Cutler, Coordinates for 6 Tbars (see Comments)

David O. H. Cutler, Coordinates for 7 Tbars (see Comments)

David O. H. Cutler, Coordinates for 8 Tbars (see Comments)

David O. H. Cutler, Coordinates for 9 Tbars (see Comments)

David O. H. Cutler, Coordinates for 10 Tbars (see Comments)

David O. H. Cutler, Coordinates for 11 Tbars (see Comments)

David O. H. Cutler, Coordinates for 12 Tbars (see Comments)

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

From Alois P. Heinz, Oct 17 2025: (Start)

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

a(n) = A058331(n) - A062781(n). (End)

For n >= 3, a(n) = A058331(n) - A001840(n-3).

Alternatively, a(n) = 2*n^2 + n + 1 - L*n + 3*L*(L-1)/2, where L = ceiling(n/3).

Maple

a:= n-> (m-> 2*n^2+1-(m*n-3*m*(m+1)/2))(floor(n/3)):
seq(a(n), n=0..59);  # Alois P. Heinz, Oct 17 2025

Mathematica

a[n_] := Module[{m = Floor[n/3]}, 2*n^2 + 1 - (m*n - 3*m*(m + 1)/2)]; Array[a, 100, 0] (* Amiram Eldar, Oct 17 2025 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {1, 3, 9, 19, 32}, 50] (* Vincenzo Librandi, Oct 20 2025 *)

Prog

(Magma) I:=[1, 3, 9, 19, 32]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Oct 20 2025

Crossrefs

A058331 is an upper bound.

Cf. A001840, A062781.

Sequence in context: A194139 A194115 A226184 * A066506 A058331 A328950

Adjacent sequences: A389611 A389612 A389613 * A389615 A389616 A389617

Keyword

nonn,easy

Author

David O. H. Cutler and N. J. A. Sloane, Oct 17 2025

Extensions

Entry updated by N. J. A. Sloane, Jan 29 2026

Status

approved