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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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)

EXAMPLE

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.

PROG

(PARI) Vec(x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5 + O(x^30)) \\ Colin Barker, May 31 2020

CROSSREFS

Cf. A255011, A331449, A334690-A334698.

Sequence in context: A261803 A184564 A184556 * A280548 A293608 A111341

Adjacent sequences:  A334694 A334695 A334696 * A334698 A334699 A334700

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified July 5 21:00 EDT 2020. Contains 335473 sequences. (Running on oeis4.)