A195321 a(n) = 18*n^2.

0, 18, 72, 162, 288, 450, 648, 882, 1152, 1458, 1800, 2178, 2592, 3042, 3528, 4050, 4608, 5202, 5832, 6498, 7200, 7938, 8712, 9522, 10368, 11250, 12168, 13122, 14112, 15138, 16200, 17298, 18432, 19602, 20808, 22050, 23328, 24642, 25992, 27378, 28800, 30258, 31752

Offset: 0

Author

Omar E. Pol, Sep 16 2011

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195316 in the same spiral.

Area of a square with diagonal 6n. - Wesley Ivan Hurt, Jun 19 2014

Number of identical tessellation tiles that are composed of 48 equilateral edge joined triangles that can be formed into a order n hexagon. The example tiles shown in the link below are tessellated with eight sphinx tiles. See A291582. - Craig Knecht, Sep 02 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Craig Knecht, Hexagon tessellation.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A195147.

Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195322, A195323.

Cf. A000290, A001105, A008600, A016766, A033428, A195160, A195316, A291582.

Programs

Magma

[18*n^2:n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

Maple

A195321:=n->18*n^2; seq(A195321(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

Mathematica

18 Range[0, 50]^2 (* or *) CoefficientList[Series[18 x*(1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{3,-3,1},{0,18,72},50] (* Harvey P. Dale, Mar 26 2023 *)

PARI

a(n)=18*n^2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 18*A000290(n) = 9*A001105(n) = 6*A033428(n) = 3*A033581(n) = 2*A016766(n).

G.f.: 18*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Jun 20 2014

From Elmo R. Oliveira, Dec 01 2024: (Start)

E.g.f.: 18*x*(1 + x)*exp(x).

a(n) = n*A008600(n) = A195147(2*n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)