A277492 Number of vertices in the metrically regular triangulation of the n-th approximation of the Koch snowflake fractal.

3, 13, 96, 780, 6684, 58812, 523932, 4693884, 42158940, 379086396, 3410401308, 30688106748, 276170940636, 2485450385340, 22368701146524, 201316901032572, 1811846472148572, 16306595700758844, 146759271112516380, 1320833079235394556

Offset: 0

Author

Dintle N Kagiso, Oct 17 2016

Keywords

• nonn

Comments

A triangulation is metrically regular if all its triangles are congruent.

Examples

			a(1) = 1+2+3+4+3 = 13 because an equilateral triangle can be cut up into 9 triangles with side length one-third which have 1+2+3+4 = 10 vertices and 3 further triangles yield 3 additional vertices.
		

Links

• Dintle N Kagiso, triangulation of snowflake
• Wikipedia, Koch snowflake
• Index entries for linear recurrences with constant coefficients, signature (13, -36).

Crossrefs

Cf. A277491.

Programs

Maple

L:=[3,13,96]: for k from 4 to 34 do: L:=[op(L),13*L[k-1]-36*L[k-2]]: od: print(L);

Mathematica

Table[(Boole[n == 0] 35/36) + (44*9^# + 21*4^#)/5 &[n - 1], {n, 0, 19}] (* or *)
CoefficientList[Series[(3 - 26 x + 35 x^2)/((1 - 4 x) (1 - 9 x)), {x, 0, 19}], x] (* Michael De Vlieger, Oct 21 2016 *)
LinearRecurrence[{13,-36},{3,13,96},30] (* Harvey P. Dale, Sep 17 2024 *)

Formula

a(n+1) = (1/5) * (44*9^n + 21*4^n) for all n > -1.

a(0)=3, a(1)=13, a(2)=96, a(n) = 13*a(n-1)-36*a(n-2) for n > 2.

G.f.: (3-26*x+35*x^2)/((1-4*x)*(1-9*x)).