A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728, 211445, 287266, 381855, 498080, 639013, 807930, 1008311, 1243840, 1518405, 1836098, 2201215, 2618256, 3091925, 3627130, 4228983, 4902800, 5654101, 6488610, 7412255, 8431168, 9551685, 10780346, 12123895, 13589280, 15183653, 16914370, 18788991

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, May 18 2020

Keywords

• nonn
• easy

Comments

An equivalent definition: Place n-1 points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of vertices in the resulting planar graph. "In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet. - Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

Equivalently, this is A334697(n) + 4*n.

This is an upper bound on A331449.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Scott R. Shannon, Colored illustration for n = 2
• Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
• Scott R. Shannon, "General position" image for n = 1.
• Scott R. Shannon, "General position" image for n = 2.
• Scott R. Shannon, "General position" image for n = 3.
• Scott R. Shannon, "General position" image for n = 4.
• Scott R. Shannon, "General position" image for n = 5.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A255011, A331449, A334690-A334698.

For the "general position" version, see also A367121 (regions), A367122 (edges), and A367117.

Programs

Mathematica

A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)
LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* Paolo Xausa, Nov 14 2023 *)

PARI

Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020

Formula

Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.

From Colin Barker, May 27 2020: (Start)

G.f.: x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5.

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

(End)

a(n+1) = A367122(n) - A367121(n) + 1 by Euler's formula.

Extensions

Edited by N. J. A. Sloane, Nov 13 2023