OFFSET
0,2
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: See "Cutting a pancake with an exotic knife".
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..3499
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.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 4, 23}, 50] (* Vincenzo Librandi, Jun 27 2025 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 26 2025
STATUS
approved