This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334698 #63 Dec 18 2024 08:09:50
%S A334698 5,58,375,1376,3685,8130,15743,27760,45621,70970,105655,151728,211445,
%T A334698 287266,381855,498080,639013,807930,1008311,1243840,1518405,1836098,
%U A334698 2201215,2618256,3091925,3627130,4228983,4902800,5654101,6488610,7412255,8431168,9551685,10780346,12123895,13589280,15183653,16914370,18788991
%N 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.
%C A334698 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
%C A334698 Equivalently, this is A334697(n) + 4*n.
%C A334698 This is an upper bound on A331449.
%H A334698 Colin Barker, <a href="/A334698/b334698.txt">Table of n, a(n) for n = 1..1000</a>
%H A334698 Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for n = 2</a>
%H A334698 Scott R. Shannon, <a href="/A334697/a334697.png">Illustration for n=3 showing interior vertices color-coded according to multiplicity.</a>
%H A334698 Scott R. Shannon, <a href="/A334698/a334698.png">"General position" image for n = 1</a>.
%H A334698 Scott R. Shannon, <a href="/A334698/a334698_1.png">"General position" image for n = 2</a>.
%H A334698 Scott R. Shannon, <a href="/A334698/a334698_2.png">"General position" image for n = 3</a>.
%H A334698 Scott R. Shannon, <a href="/A334698/a334698_3.png">"General position" image for n = 4</a>.
%H A334698 Scott R. Shannon, <a href="/A334698/a334698_4.png">"General position" image for n = 5</a>.
%H A334698 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A334698 Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.
%F A334698 From _Colin Barker_, May 27 2020: (Start)
%F A334698 G.f.: x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5.
%F A334698 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
%F A334698 (End)
%F A334698 a(n+1) = A367122(n) - A367121(n) + 1 by Euler's formula.
%t A334698 A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)
%t A334698 LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* _Paolo Xausa_, Nov 14 2023 *)
%o A334698 (PARI) Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ _Colin Barker_, May 31 2020
%Y A334698 Cf. A255011, A331449, A334690-A334698.
%Y A334698 For the "general position" version, see also A367121 (regions), A367122 (edges), and A367117.
%K A334698 nonn,easy
%O A334698 1,1
%A A334698 _Scott R. Shannon_ and _N. J. A. Sloane_, May 18 2020
%E A334698 Edited by _N. J. A. Sloane_, Nov 13 2023