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A276647
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Number of squares after the n-th generation in a symmetric (with 45-degree angles) non-overlapping Pythagoras tree.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Theorem: a(n) = 20*2^floor(n/2) + 28*2^floor((n-1)/2) - (2*n^2+10*n+33).
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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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