A249333 Number of regions formed by extending the sides of a regular n-gon.

7, 9, 16, 19, 29, 33, 46, 51, 67, 73, 92, 99, 121, 129, 154, 163, 191, 201, 232, 243, 277, 289, 326, 339, 379, 393, 436, 451, 497, 513, 562, 579, 631, 649, 704, 723, 781, 801, 862, 883, 947, 969, 1036, 1059, 1129, 1153, 1226, 1251, 1327, 1353, 1432, 1459, 1541, 1569, 1654, 1683, 1771, 1801

Offset

3,1

Comments

a(n) is the number of regions formed by the affine span of all the sides of a regular n-gon.

Links

Colin Barker, Table of n, a(n) for n = 3..1000

Philippe Ryckelynck and Laurent Smoch, On cyclotomic arrangements of lines in the plane, Int'l J. Geom. (2023) Vol. 12, No. 2, 70-88.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = (n^2+2)/2, n even, and a(n) = (n^2+n+2)/2, n odd.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - Colin Barker, Dec 14 2014

G.f.: -x^3*(3*x^4-x^3-7*x^2+2*x+7) / ((x-1)^3*(x+1)^2). - Colin Barker, Dec 14 2014

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {7, 9, 16, 19, 29}, 60] (* Harvey P. Dale, Oct 16 2019 *)

Prog

(PARI) a(n)=if(n%2, (n^2+n+2)/2, (n^2+2)/2); \\ Joerg Arndt, Dec 04 2014
(PARI) Vec(-x^3*(3*x^4-x^3-7*x^2+2*x+7)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Dec 14 2014

Crossrefs

a(n) conjecturally is the same as b(n+1) for A075855 (except for b(1), b(2), b(3)).

Sequence in context: A213220 A087680 A371303 * A020691 A366674 A145830

Adjacent sequences: A249330 A249331 A249332 * A249334 A249335 A249336

Keyword

nonn,easy

Author

Richard Stanley, Dec 03 2014

Status

approved