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.
1, 50, 363, 1360, 3665, 8106, 15715, 27728, 45585, 70930, 105611, 151680, 211393, 287210, 381795, 498016, 638945, 807858, 1008235, 1243760, 1518321, 1836010, 2201123, 2618160, 3091825, 3627026, 4228875, 4902688, 5653985, 6488490, 7412131, 8431040, 9551553, 10780210, 12123755, 13589136, 15183505, 16914218, 18788835
Offset: 1
Examples
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.
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.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Programs
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PARI
Vec(x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5 + O(x^30)) \\ 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*(1 + 45*x + 123*x^2 + 35*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)