A243099 A002061 and A000217 interleaved.

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, 133, 78, 157, 91, 183, 105, 211, 120, 241, 136, 273, 153, 307, 171, 343, 190, 381, 210, 421, 231, 463, 253, 507, 276, 553, 300, 601

Offset

3,6

Comments

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.

Links

Paolo Xausa, Table of n, a(n) for n = 3..10000

Kival Ngaokrajang, Illustration of initial terms.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

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.

From Colin Barker, Aug 19 2014: (Start)

a(n) = (71+41*(-1)^n-4*(7+3*(-1)^n)*n+(3+(-1)^n)*n^2)/16 for n>4.

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>10.

G.f.: -x^6*(x^4+x+1) / ((x-1)^3*(x+1)^3). (End)

Sum_{n>=6} (-1)^(n+1)/a(n) = 2 - tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - Amiram Eldar, Feb 11 2024

Mathematica

With[{r=Range[50]}, Join[{0, 0, 0}, Riffle[r^2-r+1, PolygonalNumber[r]]]] (* or *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 0, 0, 1, 1, 3, 3, 7}, 100] (* Paolo Xausa, Feb 09 2024 *)

Prog

(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))
for (n=3, 100, print1(a(n), ", "))
(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

Crossrefs

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.

Sequence in context: A096273 A069981 A000199 * A324877 A359947 A201932

Adjacent sequences: A243096 A243097 A243098 * A243100 A243101 A243102

Keyword

nonn,easy

Author

Kival Ngaokrajang, Aug 19 2014

Status

approved