A367015 Number of regions formed after n points have been placed in general position on each edge of a triangle (as in A365929).

1, 4, 28, 136, 445, 1126, 2404, 4558, 7921, 12880, 19876, 29404, 42013, 58306, 78940, 104626, 136129, 174268, 219916, 274000, 337501, 411454, 496948, 595126, 707185, 834376, 978004, 1139428, 1320061, 1521370, 1744876, 1992154, 2264833, 2564596, 2893180, 3252376, 3644029, 4070038

Offset

0,2

Comments

See A365929 for more information.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Scott R. Shannon, Image for n = 2.

Scott R. Shannon, Image for n = 3. Note that although the number of k-gons will vary as the edge points change position the total number of regions will stay constant (at 136 for n = 3) as long as all internal vertices remain simple.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

Conjecture: a(n) = (9*n^4 - 12*n^3 + 15*n^2 + 4)/4. [This is now a theorem - N. J. A. Sloane, Dec 31 2025]

a(n) = A366932(n) - 3*A366478(n) + 1 by Euler's formula.

From Elmo R. Oliveira, Apr 17 2026: (Start)

G.f.: (1 - x + 18*x^2 + 26*x^3 + 10*x^4)/(1 - x)^5.

E.g.f.: (1/4)*(4 + 12*x + 42*x^2 + 42*x^3 + 9*x^4)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

Mathematica

A367015[n_] := 3*n^2*(n*(3*n - 4) + 5)/4 + 1; Array[A367015, 50, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 4, 28, 136, 445}, 50] (* Paolo Xausa, Jan 02 2026 *)

Crossrefs

Cf. A365929 (internal vertices), A366932 (edges), A366478 (vertices/3).

Sequence in context: A270275 A296638 A270721 * A377885 A270892 A271603

Adjacent sequences: A367012 A367013 A367014 * A367016 A367017 A367018

Keyword

nonn,easy

Author

Scott R. Shannon, Nov 01 2023

Status

approved