A176646 a(n) is the number of convex pentagons in an n-triangular net.

0, 0, 3, 21, 78, 216, 498, 1014, 1884, 3264, 5349, 8379, 12642, 18480, 26292, 36540, 49752, 66528, 87543, 113553, 145398, 184008, 230406, 285714, 351156, 428064, 517881, 622167, 742602, 880992, 1039272, 1219512, 1423920, 1654848, 1914795, 2206413, 2532510, 2896056

Offset

1,3

Comments

See P(n) in Theorem 2.1, p.2 of Zhu.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Jun-Ming Zhu, The number of convex pentagons and hexagons in an n-triangular net, arXiv:1012.4058 [math.CO], 2010. See P(n), formula (1), page 2.

Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Formula

From G. C. Greubel, Jul 03 2021: (Start)

a(n) = (1/320)*(12*n^5 - 10*n^4 - 60*n^3 + 40*n^2 + 48*n - 15 + 15*(-1)^n).

a(2*n+1) = n*(n+1)*(12*n^3 + 13*n^2 - 8*n - 2)/10.

a(2*n) = n*(4*n-3)*(3*n+1)*(n-1)*(n+1)/10.

G.f.: 3*x^3*(1 + 2*x)/((1 + x)*(1 - x)^6).

E.g.f.: (1/320)*(15*exp(-x) - (15 -30*x +30*x^2 -180*x^3 -110*x^4 -12*x^5)*exp(x)). (End)

a(n) = 3*A001753(n-3) + 6*A001753(n-4). - R. J. Mathar, Jul 04 2021

Maple

A176646:= n-> (12*n^5 -10*n^4 -60*n^3 +40*n^2 +48*n -15 +15*(-1)^n)/320;
seq(A176646(n), n=1..40); # R. J. Mathar, Dec 21 2010

Mathematica

LinearRecurrence[{5, -9, 5, 5, -9, 5, -1}, {0, 0, 3, 21, 78, 216, 498}, 40] (* Harvey P. Dale, Jan 14 2015 *)

Prog

(Magma) [(1/320)*(12*n^5 -10*n^4 -60*n^3 +40*n^2 +48*n -15 +15*(-1)^n): n in [1..40]]; // G. C. Greubel, Jul 02 2021
(Sage) [(1/320)*(12*n^5 -10*n^4 -60*n^3 +40*n^2 +48*n -15 +15*(-1)^n) for n in (1..40)] # G. C. Greubel, Jul 02 2021
(PARI) f(k) = (12*k^5 + 25*k^4 + 5*k^3 - 10*k^2 - 2*k)/10;
g(k) = (12*k^5 - 5*k^4 - 15*k^3 + 5*k^2 + 3*k)/10;
a(n) = if (n%2, f((n-1)/2), g(n/2)); \\ Michel Marcus, Jul 04 2021

Crossrefs

Cf. A166189 (for the hexagons).

Sequence in context: A109721 A067002 A110450 * A102832 A112851 A253943

Adjacent sequences: A176643 A176644 A176645 * A176647 A176648 A176649

Keyword

nonn,easy

Author

Jonathan Vos Post, Dec 21 2010

Extensions

Definition corrected and edited by Michel Marcus and G. C. Greubel, Jul 03 2021

Status

approved