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A276647 Number of squares after the n-th generation in a symmetric (with 45 degree angles) non-overlapping Pythagoras tree. 2
1, 3, 7, 15, 31, 59, 107, 183, 303, 483, 755, 1151, 1735, 2571, 3787, 5511, 7999, 11507, 16547, 23631, 33783, 48027, 68411, 96983, 137839, 195075, 276883, 391455, 555175, 784427, 1111979, 1570599, 2225823, 3143187, 4453763, 6288623, 8909911, 12579771 (list; graph; refs; listen; history; text; internal format)
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

Partial sums of A276677.

Sequence in context: A151338 A229006 A023424 * A006778 A007574 A034480

Adjacent sequences:  A276644 A276645 A276646 * A276648 A276649 A276650

KEYWORD

nonn,easy

AUTHOR

Ernst van de Kerkhof, Sep 13 2016

STATUS

approved

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Last modified October 6 07:09 EDT 2022. Contains 357262 sequences. (Running on oeis4.)