OFFSET
1,2
COMMENTS
A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hex Number.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = binomial(A003215(n-1), 3)
= binomial(3*n^2-3*n+1, 3)
= 1/2*n*(n-1)*(3*n^2-3*n+1)*(3*n^2-3*n-1)
= 9/2*n^6-27/2*n^5+27/2*n^4-9/2*n^3-1/2*n^2+1/2*n.
G.f.: -x^2*(35*x^4+724*x^3+1722*x^2+724*x+35) / (x-1)^7. - Colin Barker, Apr 18 2014
Sum_{n>=2} 1/a(n) = sqrt(3/7)*Pi*tan(sqrt(7/3)*Pi/2) + sqrt(3)*Pi*tanh(Pi/(2*sqrt(3))) - 2. - Amiram Eldar, Feb 17 2024
MAPLE
seq(binomial(3*n^2-3*n+1, 3), n=1..28); # Martin Renner, May 31 2014
op(PolynomialTools[CoefficientList](convert(series(-x^2*(35*x^4+724*x^3+1722*x^2+724*x+35)/(x-1)^7, x=0, 29), polynom), x)[2..29]); # Martin Renner, May 31 2014
MATHEMATICA
CoefficientList[Series[- x(35 x^4 + 724 x^3 + 1722 x^2 + 724 x + 35)/(x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 19 2014 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 35, 969, 7770, 35990, 121485, 333375}, 40] (* Harvey P. Dale, Sep 12 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Martin Renner, Apr 17 2014
STATUS
approved