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A334697 a(n) is the number of interior points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), counted with multiplicity.

%I A334697 #31 May 31 2020 06:22:54
%S A334697 1,50,363,1360,3665,8106,15715,27728,45585,70930,105611,151680,211393,
%T A334697 287210,381795,498016,638945,807858,1008235,1243760,1518321,1836010,
%U A334697 2201123,2618160,3091825,3627026,4228875,4902688,5653985,6488490,7412131,8431040,9551553,10780210,12123755,13589136,15183505,16914218,18788835
%N A334697 a(n) is the number of interior points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), counted with multiplicity.
%H A334697 Colin Barker, <a href="/A334697/b334697.txt">Table of n, a(n) for n = 1..1000</a>
%H A334697 Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for n = 2</a>
%H A334697 Scott R. Shannon, <a href="/A334697/a334697.png">Illustration for n=3 showing interior vertices color-coded according to multiplicity.</a>
%H A334697 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A334697 Theorem: a(n) = n*(17*n^3-30*n^2+19*n-4)/2.
%F A334697 From _Colin Barker_, May 27 2020: (Start)
%F A334697 G.f.: x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5.
%F A334697 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 A334697 (End)
%e A334697 Scott Shannon's illustration for n=2 shows 29 interior intersection points, of which 20 are simple intersections, 8 are triple intersections, and one (the central point) is a 4-fold intersection. A point where d lines meet is equivalent to C(d,2) simple points. So a(2) = 20*1 + 8*3 + 1*6 = 50.
%o A334697 (PARI) Vec(x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5 + O(x^30)) \\ _Colin Barker_, May 31 2020
%Y A334697 Cf. A255011, A331449, A334690-A334698.
%K A334697 nonn,easy
%O A334697 1,2
%A A334697 _Scott R. Shannon_ and _N. J. A. Sloane_, May 18 2020