A241219 -id:A241219 - cp's OEIS frontend

A240826 Number of ways to choose three points on a centered hexagonal grid of size n.

0, 35, 969, 7770, 35990, 121485, 333375, 790244, 1679580, 3280455, 5989445, 10349790, 17083794, 27128465, 41674395, 62207880, 90556280, 128936619, 180007425, 246923810, 333395790, 443749845, 582993719, 756884460, 971999700, 1235812175, 1556767485, 1944365094

Offset: 1

Martin Renner, Apr 17 2014

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

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).

Cf. A003215, A178208, A241219.

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

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 *)

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

A241220 Number of lines through at least two points of a centered hexagonal grid of size n.

Original entry on oeis.org

0, 15, 84, 312, 852, 1878, 3654, 6546, 10680, 16668, 25002, 35910, 50136, 68190, 90462, 118200, 152274, 192828, 240480, 296880, 361962, 437832, 525756, 625440, 739146, 867864, 1011822, 1174062, 1354572, 1554114, 1775568, 2020848, 2289054, 2582760, 2905410

Offset: 1

Martin Renner, Apr 17 2014

nonn

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Hex Number.

Cf. A018808, A241219, A003215.

c(n,s,fmin,fmax)={sum(k=1+s, n, max(0, fmax(k-s)-max(fmin(k)-1,if(k-2*s>0,fmax(k-2*s)))))}
b(n, u, v)={c(2*n-1, u, i->max(0,i-n)+1+i\u*v, i->min(i,n)+n-1+i\u*v)}
a(n)={3*((n>1)*(2*n-1) + sum(u=1, 2*n-3, sum(v=1, 2*n-2-u, if(gcd(u,v)==1, b(n,u,v), 0))))} \\ Andrew Howroyd, Sep 18 2017

a(15)-a(16) from Martin Renner, Apr 27 2014

Terms a(17) and beyond from Andrew Howroyd, Sep 18 2017

cp's OEIS Frontend

A240826 Number of ways to choose three points on a centered hexagonal grid of size n.

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