A282563 One third of the number of edges in the metrically regular triangulation of the n-th approximation of the Koch snowflake fractal.

1, 8, 68, 596, 5300, 47444, 425972, 3829652, 34450484, 309988820, 2789637236, 25105686548, 225946984628, 2033506084436, 18301487651060, 164713120424084, 1482417010074932, 13341748795707092, 120075721981494644, 1080681429113975060

Offset: 1

Author

Dintle N Kagiso, Feb 18 2017

Keywords

• nonn
• easy

Comments

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

Three times the n-th sequence entry a(n) is the number of edges of the n-th approximation of the Koch snowflake fractal.

Examples

			a(1)=1, and there are three edges in a triangle. a(2)=8 and there are 24 edges in the second approximation of the Koch fractal.
		

Links

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

Crossrefs

Cf. A277491, A277492.

Programs

Maple

L:=[1,8]: for k from 3 to 30 do: L:=[op(L),13*L[k-1]-36*L[k-2]]: od: print(L);

Mathematica

CoefficientList[Series[(1 - 5 x)/((1 - 4 x) (1 - 9 x)), {x, 0, 19}], x] (* or *) Table[(1/5) (4*9^# + 4^#) &[n + 1], {n, -1, 19}] (* Michael De Vlieger, Feb 18 2017 *)
LinearRecurrence[{13,-36},{1,8},30] (* Harvey P. Dale, Sep 22 2019 *)

Formula

a(n+1) = (1/5) * (4*9^n +4^n) for all n > 0.

a(1)=1, a(2)=8, a(3)=68, a(n) = 13*a(n-1)-36*a(n-2) for n > 2.

G.f.: (1-5*x)/((1-4*x)*(1-9*x)).