OFFSET
1,1
COMMENTS
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
Equivalently, this is A334697(n) + 4*n.
This is an upper bound on A331449.
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.
Scott R. Shannon, "General position" image for n = 1.
Scott R. Shannon, "General position" image for n = 2.
Scott R. Shannon, "General position" image for n = 3.
Scott R. Shannon, "General position" image for n = 4.
Scott R. Shannon, "General position" image for n = 5.
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*(5 + 33*x + 135*x^2 + 31*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)
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, {5, 58, 375, 1376, 3685}, 50] (* Paolo Xausa, Nov 14 2023 *)
PROG
(PARI) Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Scott R. Shannon and N. J. A. Sloane, May 18 2020
EXTENSIONS
Edited by N. J. A. Sloane, Nov 13 2023
STATUS
approved