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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.
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%I #60 Nov 14 2023 16:21:41

%S 5,58,375,1376,3685,8130,15743,27760,45621,70970,105655,151728,211445,

%T 287266,381855,498080,639013,807930,1008311,1243840,1518405,1836098,

%U 2201215,2618256,3091925,3627130,4228983,4902800,5654101,6488610,7412255,8431168,9551685,10780346,12123895,13589280,15183653,16914370,18788991

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

%C This is an upper bound on A331449.

%H Colin Barker, <a href="/A334698/b334698.txt">Table of n, a(n) for n = 1..1000</a>

%H Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for n = 2</a>

%H Scott R. Shannon, <a href="/A334697/a334697.png">Illustration for n=3 showing interior vertices color-coded according to multiplicity.</a>

%H Scott R. Shannon, <a href="/A334698/a334698.png">"General position" image for n = 1</a>.

%H Scott R. Shannon, <a href="/A334698/a334698_1.png">"General position" image for n = 2</a>.

%H Scott R. Shannon, <a href="/A334698/a334698_2.png">"General position" image for n = 3</a>.

%H Scott R. Shannon, <a href="/A334698/a334698_3.png">"General position" image for n = 4</a>.

%H Scott R. Shannon, <a href="/A334698/a334698_4.png">"General position" image for n = 5</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F From _Colin Barker_, May 27 2020: (Start)

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

%F 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 (End)

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

%t A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)

%t LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* _Paolo Xausa_, Nov 14 2023 *)

%o (PARI) Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ _Colin Barker_, May 31 2020

%Y Cf. A255011, A331449, A334690-A334698.

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

%K nonn,easy

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, May 18 2020

%E Edited by _N. J. A. Sloane_, Nov 13 2023