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
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Aug 19 2014
STATUS
approved