A239571 Number of ways to place 5 points on a triangular grid of side n so that no two of them are adjacent.

0, 0, 27, 999, 11565, 74811, 342042, 1239525, 3799488, 10259640, 25076952, 56552364, 119324403, 238062357, 452774595, 826245798, 1454229216, 2479147536, 4108199481, 6636929805, 10479498849, 16207085223, 24596072424, 36687908235, 53862785520, 77929575480

Offset

3,3

Comments

Rotations and reflections of placements are counted. If they are to be ignored see A239575.

a(n) is the coefficient of x^5 in the independence polynomial of the (n-1)-triangular grid graph. - Eric W. Weisstein, Mar 27 2026

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..1000

Eric Weisstein's World of Mathematics, Independence Polynomial.

Eric Weisstein's World of Mathematics, Triangular Grid Graph.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1)

Formula

a(n) = (n -3) * (n -4) * (n^8 +12*n^7 -58*n^6 -860*n^5 +2141*n^4 +23728*n^3 -61316*n^2 -244928*n +770880)/3840.

G.f.: -3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9) / (x-1)^11. - Colin Barker, Mar 22 2014

a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Eric W. Weisstein, Mar 27 2026

Mathematica

CoefficientList[Series[-3 x^2 (40 x^8 - 185 x^7 + 198 x^6 + 213 x^5 - 243 x^4 - 638 x^3 + 687 x^2 + 234 x + 9)/(x - 1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)
Table[(n - 4) (n - 3) (n^8 + 12 n^7 - 58 n^6 - 860 n^5 + 2141 n^4 + 23728 n^3 - 61316 n^2 - 244928 n + 770880)/3840, {n, 3, 20}] (* Eric W. Weisstein, Mar 27 2026 *)
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 27, 999, 11565, 74811, 342042, 1239525, 3799488, 10259640, 25076952}, 20] (* Eric W. Weisstein, Mar 27 2026 *)

Prog

(PARI) concat([0, 0], Vec(-3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9)/(x-1)^11 + O(x^100))) \\ Colin Barker, Mar 22 2014
(Magma) [(n^2-7*n+12)*(n^8+12*n^7-58*n^6-860*n^5+2141*n^4 +23728*n^3-61316*n^2-244928*n+770880)/3840: n in [3..40]]; // Vincenzo Librandi, Mar 23 2014

Crossrefs

Cf. A239567, A239575, A239568 (2 points), A239569 (3 points), A239570 (4 points), A282998 (6 points).

Sequence in context: A129999 A132059 A292362 * A017019 A143366 A143705

Adjacent sequences: A239568 A239569 A239570 * A239572 A239573 A239574

Keyword

nonn,easy

Author

Heinrich Ludwig, Mar 22 2014

Status

approved