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A249333 Number of regions formed by extending the sides of a regular n-gon. 1
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 (list; graph; refs; listen; history; text; internal format)
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
Philippe Ryckelynck and Laurent Smoch, On cyclotomic arrangements of lines in the plane, Int'l J. Geom. (2023) Vol. 12, No. 2, 70-88.
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: A158891 A213220 A087680 * A020691 A366674 A145830
KEYWORD
nonn,easy
AUTHOR
Richard Stanley, Dec 03 2014
STATUS
approved

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Last modified February 21 12:33 EST 2024. Contains 370235 sequences. (Running on oeis4.)