OFFSET
0,2
COMMENTS
Non-overlapping is to be understood as: any two different squares in the tree can never share more than one side, disallowing area overlap. In branches where an area overlap is about to occur, growth is terminated.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Ernst van de Kerkhof, Illustration of a(6)
"QuantumKiwi", A Year in the Life of a Pythagoras Tree, YouTube, (2008).
Wikipedia, Pythagoras tree (fractal)
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,6,-2).
FORMULA
Theorem: a(n) = 20*2^floor(n/2) + 28*2^floor((n-1)/2) - (2*n^2+10*n+33).
From Colin Barker, Sep 20 2016: (Start)
G.f.: (1+x)^2*(1-2*x+2*x^2) / ((1-x)^3*(1-2*x^2)).
a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+6*a(n-4)-2*a(n-5) for n>4.
a(n) = (-25+2^((n-1)/2)*(24-24*(-1)^n+17*sqrt(2)+17*(-1)^n*sqrt(2))-4*(1+n)-2*(1+n)*(2+n)). Therefore:
a(n) = 17*2^(n/2+1)-2*n^2-10*n-33 for n even.
a(n) = 3*2^((n+7)/2)-2*n^2-10*n-33 for n odd. (End)
MATHEMATICA
TableForm[Table[{n, 20 * 2^Floor[n/2] + 28*2^Floor[(n-1)/2] - (2n^2 + 10n + 33)}, {n, 0, 100, 1}], TableSpacing -> {1, 5}]
LinearRecurrence[{3, -1, -5, 6, -2}, {1, 3, 7, 15, 31}, 50] (* Harvey P. Dale, May 07 2019 *)
PROG
(PARI) Vec((1+x)^2*(1-2*x+2*x^2)/((1-x)^3*(1-2*x^2)) + O(x^50)) \\ Colin Barker, Sep 20 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ernst van de Kerkhof, Sep 13 2016
STATUS
approved