A342914 Number of grid points covered by a truncated triangle drawn on the hexagonal lattice with the short sides having length n and the long sides length 2*n.

1, 12, 36, 73, 123, 186, 262, 351, 453, 568, 696, 837, 991, 1158, 1338, 1531, 1737, 1956, 2188, 2433, 2691, 2962, 3246, 3543, 3853, 4176, 4512, 4861, 5223, 5598, 5986, 6387, 6801, 7228, 7668, 8121, 8587, 9066, 9558, 10063, 10581, 11112, 11656, 12213, 12783, 13366

Offset

0,2

Comments

The shapes can be constructed using compass and straightedge. a(n) identical circles must be drawn to create a truncated triangle whose shortest side is n radius lengths. See illustrations of the initial terms in the links.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Mert Aydemir, Semi-postesque hexagon lenis

Mert Aydemir, Illustration of n=1

Mert Aydemir, Illustration of n=3

Mert Aydemir, Illustration of n=4

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = (13*n^2 + 9*n + 2)/2.

a(n) = A000217(4*n+1) - 3*A000217(n). - Andrew Howroyd, Apr 01 2021

G.f.: (1 + 9*x + 3*x^2)/(1 - x)^3. - Stefano Spezia, Apr 01 2021

Example

a(1) = 12,        a(2) = 36:
       * *           * * *
      * * *         * * * *
     * * * *       * * * * *
      * * *       * * * * * *
                 * * * * * * *
                  * * * * * *
                   * * * * *

Mathematica

CoefficientList[Series[(1+9x+3x^2)/(1-x)^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3, -3, 1}, {1, 12, 36}, 50] (* Harvey P. Dale, Apr 08 2023 *)

Prog

(PARI) a(n) = (13*n^2+9*n+2)/2 \\ Andrew Howroyd, Apr 01 2021

Crossrefs

Cf. A000217, A003215 (regular hexagon).

Sequence in context: A043920 A371912 A049598 * A152135 A080562 A212963

Adjacent sequences: A342911 A342912 A342913 * A342915 A342916 A342917

Keyword

nonn,easy

Author

Mert Aydemir, Mar 31 2021

Status

approved