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%I #25 Apr 11 2023 16:40:23
%S 7,9,16,19,29,33,46,51,67,73,92,99,121,129,154,163,191,201,232,243,
%T 277,289,326,339,379,393,436,451,497,513,562,579,631,649,704,723,781,
%U 801,862,883,947,969,1036,1059,1129,1153,1226,1251,1327,1353,1432,1459,1541,1569,1654,1683,1771,1801
%N Number of regions formed by extending the sides of a regular n-gon.
%C a(n) is the number of regions formed by the affine span of all the sides of a regular n-gon.
%H Colin Barker, <a href="/A249333/b249333.txt">Table of n, a(n) for n = 3..1000</a>
%H Philippe Ryckelynck and Laurent Smoch, <a href="https://ijgeometry.com/product/philippe-ryckelynck-and-laurent-smoch-on-cyclotomic-arrangements-of-lines-in-the-plane/">On cyclotomic arrangements of lines in the plane</a>, Int'l J. Geom. (2023) Vol. 12, No. 2, 70-88.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = (n^2+2)/2, n even, and a(n) = (n^2+n+2)/2, n odd.
%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - _Colin Barker_, Dec 14 2014
%F 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
%t LinearRecurrence[{1,2,-2,-1,1},{7,9,16,19,29},60] (* _Harvey P. Dale_, Oct 16 2019 *)
%o (PARI) a(n)=if(n%2,(n^2+n+2)/2,(n^2+2)/2); \\ _Joerg Arndt_, Dec 04 2014
%o (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
%Y a(n) conjecturally is the same as b(n+1) for A075855 (except for b(1), b(2), b(3)).
%K nonn,easy
%O 3,1
%A _Richard Stanley_, Dec 03 2014
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