A069813 Maximum number of triangles in polyiamond with perimeter n.

1, 2, 3, 6, 7, 10, 13, 16, 19, 24, 27, 32, 37, 42, 47, 54, 59, 66, 73, 80, 87, 96, 103, 112, 121, 130, 139, 150, 159, 170, 181, 192, 203, 216, 227, 240, 253, 266, 279, 294, 307, 322, 337, 352, 367, 384, 399, 416, 433, 450, 467, 486, 503, 522, 541, 560, 579

Offset

3,2

Links

Colin Barker, Table of n, a(n) for n = 3..1000

W. C. Yang, R. R. Meyer, Maximal and minimal polyiamonds, 2002.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Formula

a(n) = round(n^2/6) - (0 if n = 0 mod 6, 1 else) = A056829(n)-A097325(n).

From Colin Barker, Jan 18 2015: (Start)

a(n) = round((-25 + 9*(-1)^n + 8*exp(-2/3*i*n*Pi) + 8*exp((2*i*n*Pi)/3) + 6*n^2)/36), where i=sqrt(-1).

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

a(n) = A001399(n-3) + A001399(n-4) + A001399(n-6) - A001399(n-7). - R. J. Mathar, Jul 14 2015

Example

a(10) = 16 because the maximum number of triangles in a polyiamond of perimeter 10 is 16.

Maple

A069813 := proc(n)
    round(n^2/6) ;
    if modp(n, 6) <> 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, Jul 14 2015

Mathematica

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 2, 3, 6, 7, 10}, 60] (* Jean-François Alcover, Jan 03 2020 *)

Prog

(PARI) a(n) = round(n^2/6) - (n % 6 != 0) \\ Michel Marcus, Jul 17 2013
(PARI) Vec(x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Jan 19 2015
(Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)))); // Marius A. Burtea, Jan 03 2020

Crossrefs

Cf. A000105, A000577, A027709, A030511 (bisection), A057729, A067628.

Sequence in context: A331075 A378595 A062720 * A266534 A293392 A161336

Adjacent sequences: A069810 A069811 A069812 * A069814 A069815 A069816

Keyword

nonn,easy

Author

Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

Status

approved