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%I #28 Feb 11 2024 02:40:47
%S 0,0,0,1,1,3,3,7,6,13,10,21,15,31,21,43,28,57,36,73,45,91,55,111,66,
%T 133,78,157,91,183,105,211,120,241,136,273,153,307,171,343,190,381,
%U 210,421,231,463,253,507,276,553,300,601
%N A002061 and A000217 interleaved.
%C a(n) is the number of the distinct symmetric 6-gon in a regular n-gon where vertices of 6-gon are placed on vertices of n-gon. See illustration.
%H Paolo Xausa, <a href="/A243099/b243099.txt">Table of n, a(n) for n = 3..10000</a>
%H Kival Ngaokrajang, <a href="/A243099/a243099.pdf">Illustration of initial terms</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F a(3) = a(4) = a(5) = 0; for n >= 6, a(n) = (n/2-2)^2-(n/2-2)+1 if even n, a(n) = (n/2-5/2)*(n/2-5/2+1)/2 if odd n.
%F From _Colin Barker_, Aug 19 2014: (Start)
%F a(n) = (71+41*(-1)^n-4*(7+3*(-1)^n)*n+(3+(-1)^n)*n^2)/16 for n>4.
%F a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>10.
%F G.f.: -x^6*(x^4+x+1) / ((x-1)^3*(x+1)^3). (End)
%F Sum_{n>=6} (-1)^(n+1)/a(n) = 2 - tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - _Amiram Eldar_, Feb 11 2024
%t With[{r=Range[50]}, Join[{0, 0, 0}, Riffle[r^2-r+1, PolygonalNumber[r]]]] (* or *)
%t LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 0, 0, 1, 1, 3, 3, 7}, 100] (* _Paolo Xausa_, Feb 09 2024 *)
%o (PARI)a(n) = if(n<6,0,if(Mod(n,2)==0,(n/2-2)^2-(n/2-2)+1,(n/2-5/2)*(n/2-5/2+1)/2))
%o for (n=3,100,print1(a(n),", "))
%o (PARI) concat([0,0,0], Vec(-x^6*(x^4+x+1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ _Colin Barker_, Aug 19 2014
%Y Cf. A001399: 3-gon in n-gon, A226088: 4-gon in n-gon, A004526: symmetric 4-gon in n-gon, A008805: symmetric 5-gon in n-gon.
%K nonn,easy
%O 3,6
%A _Kival Ngaokrajang_, Aug 19 2014
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