A166189 Number of 3 X 3 X 3 triangular nonnegative integer arrays with all sums of an element and its neighbors <= n.

1, 7, 29, 90, 232, 524, 1072, 2030, 3613, 6111, 9905, 15484, 23464, 34608, 49848, 70308, 97329, 132495, 177661, 234982, 306944, 396396, 506584, 641186, 804349, 1000727, 1235521, 1514520, 1844144, 2231488, 2684368, 3211368

Offset: 0

Author

R. H. Hardin, Oct 09 2009

Keywords

• nonn
• easy

Comments

a(n) gives the number of hexagons that have vertices at the lattice points and sides on lattice lines of a triangular lattice with sides n+3. Note that the hexagons can be non-regular. This problem appeared as ConvexHexagons in Single Round Match 455 in TopCoder. - Dmitry Kamenetsky, Dec 17 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jun-Ming Zhu, The number of convex pentagons and hexagons in an n-triangular net, arXiv:1012.4058 [math.CO], 2010; See H(n), formula 3, on page 4.
• Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Crossrefs

Cf. A176646, A011888 (first differences).

Programs

Magma

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1-x^3)/((1-x^2)*(1-x)^7) )); // G. C. Greubel, Jul 02 2021

Mathematica

LinearRecurrence[{6,-14,14,0,-14,14,-6,1}, {1,7,29,90,232,524,1072,2030}, 51] (* G. C. Greubel, Jul 02 2021 *)

PARI

\\ using Zhu expressions
f(k) = (8*k^6 + 24*k^5 + 25*k^4 + 10*k^3 - 3*k^2 -4*k)/60;
g(k) = (8*k^6 - 5*k^4 - 3*k^2)/60;
a(n) = n+=3; if (n%2, f((n-1)/2), g(n/2)); \\ Michel Marcus, Jul 04 2021

Sage

def a(n): return (n+2)*(n+4)*(2*n^4 +24*n^3 +105*n^2 +198*n +120)/960 if (n%2==0) else (n+1)*(n+3)^2*(n+5)*(2*n*(n+6) +21)/960
[a(n) for n in (0..50)] # G. C. Greubel, Jul 02 2021

Formula

From G. C. Greubel, Jul 02 2021: (Start)

a(n) = (1/1920)*(4*n^6 +72*n^5 +530*n^4 +2040*n^3 +4296*n^2 +4608*n +1905 +15*(-1)^n).

a(2*n+1) = (1/15)*binomial(n+2, 2)*binomial(n+3, 2)*(8*n^2 + 32*n + 35).

a(2*n) = (1/30)*binomial(n+2, 2)*(8*n^4 + 48*n^3 + 105*n^2 + 99*n + 30).

G.f.: (1 - x^3)/((1-x^2)*(1-x)^7).

E.g.f.: (1/1920)*((1905 +2*x*(5775 +7665*x +3690*x^2 +755*x^3 +66*x^4 +2*x^5))*exp(x) + 15*exp(-x)). (End)

a(n) = A001779(n)-A001779(n-3). - R. J. Mathar, Jul 04 2021