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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
Ernst van de Kerkhof, Illustration of a(6)
"QuantumKiwi", A Year in the Life of a Pythagoras Tree, YouTube, (2008).
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 29 02:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)