

A334697


a(n) is the number of interior points in the nth 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 colorcoded according to multiplicity.
Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).


FORMULA

Theorem: a(n) = n*(17*n^330*n^2+19*n4)/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(n1)  10*a(n2) + 10*a(n3)  5*a(n4) + a(n5) 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 4fold 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, A334690A334698.
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



