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A247618 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.)

%I A247618 #21 Feb 08 2024 09:46:48
%S A247618 1,5,17,45,105,229,481,989,2009,4053,8145,16333,32713,65477,131009,
%T A247618 262077,524217,1048501,2097073,4194221,8388521,16777125,33554337,
%U A247618 67108765,134217625,268435349,536870801,1073741709,2147483529,4294967173,8589934465
%N A247618 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.)
%C A247618 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.
%H A247618 Kival Ngaokrajang, <a href="/A247618/a247618_1.pdf">Illustration of initial terms</a>
%H A247618 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A247618 a(0) = 1, for n >= 1, a(n) = 4*A000225(n) + a(n-1).
%F A247618 From _Colin Barker_, Sep 21 2014: (Start)
%F A247618 a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3).
%F A247618 a(n) = (-7+2^(3+n)-4*n).
%F A247618 G.f.: -(2*x^2+x+1) / ((x-1)^2*(2*x-1)).
%F A247618 (End)
%o A247618 (PARI)
%o A247618 a(n) = if (n<1,1,4*(2^n-1)+a(n-1))
%o A247618 for (n=0,50,print1(a(n),", "))
%o A247618 (PARI)
%o A247618 Vec(-(2*x^2+x+1) / ((x-1)^2*(2*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 21 2014
%Y A247618 Cf. A000225, A061777, A001844, A247619, A247620.
%K A247618 nonn,easy
%O A247618 0,2
%A A247618 _Kival Ngaokrajang_, Sep 20 2014