OFFSET
1,2
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.
LINKS
Dintle N Kagiso, triangulation of snowflake
Wikipedia, Koch snowflake
Index entries for linear recurrences with constant coefficients, signature (13,-36).
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)).
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dintle N Kagiso, Feb 18 2017
STATUS
approved