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A247618
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Start with a single square; at n-th generation add a square at each expandable vertex; a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)
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12
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1, 5, 17, 45, 105, 229, 481, 989, 2009, 4053, 8145, 16333, 32713, 65477, 131009, 262077, 524217, 1048501, 2097073, 4194221, 8388521, 16777125, 33554337, 67108765, 134217625, 268435349, 536870801, 1073741709, 2147483529, 4294967173, 8589934465
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OFFSET
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0,2
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COMMENTS
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Inspired by A061777, let us assign label "1" to an origin square; at n-th generation add a square at each expandable vertex, i.e. a vertex such that the new added generations will not overlap to the existing ones, but overlapping among new generations are allowed. The non-overlapping squares will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. The squares count is A001844. See illustration. For n >= 1, (a(n) - a(n-1))/4 is A000225.
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LINKS
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FORMULA
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a(0) = 1, for n >= 1, a(n) = 4*A000225(n) + a(n-1).
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3).
a(n) = (-7+2^(3+n)-4*n).
G.f.: -(2*x^2+x+1) / ((x-1)^2*(2*x-1)).
(End)
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PROG
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(PARI)
a(n) = if (n<1, 1, 4*(2^n-1)+a(n-1))
for (n=0, 50, print1(a(n), ", "))
(PARI)
Vec(-(2*x^2+x+1) / ((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 21 2014
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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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