A383464 a(n) = 8*n^2 - 5*n + 1.

1, 4, 23, 58, 109, 176, 259, 358, 473, 604, 751, 914, 1093, 1288, 1499, 1726, 1969, 2228, 2503, 2794, 3101, 3424, 3763, 4118, 4489, 4876, 5279, 5698, 6133, 6584, 7051, 7534, 8033, 8548, 9079, 9626, 10189, 10768, 11363, 11974, 12601, 13244, 13903, 14578, 15269

Offset: 0

Author

N. J. A. Sloane, Jun 26 2025

Keywords

• nonn
• easy

Comments

This is equal to A139272(n) + 1, but has its own entry because of an important geometrical interpretation.

Definition: A k-legged Wu is a pencil of k semi-infinite lines originating from a common point.

A 2-legged Wu is a long-legged V (see A130883), and a 3-legged Wu is a long-legged Wu as in A140064.

Theorem (David Cutler, Jonathan Pei, and Edward Xiong, Jun 24 2025): a(n) is the maximum number of regions in the plane that can be formed from n copies of a 4-legged Wu.

Proof: [To be added]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3499
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A130083, A139272, A140064.

Programs

Magma

I:=[1, 4, 23]; [n le 3 select I[n] else 3*Self(n-1)-3* Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 27 2025

Mathematica

LinearRecurrence[{3,-3,1},{1,4,23},50] (* Vincenzo Librandi, Jun 27 2025 *)

Formula

G.f.: (1 + x + 14*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(1 + 3*x + 8*x^2). - Stefano Spezia, Jun 30 2025