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A247903 Start with a single square; at n-th generation add a square at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

%I A247903 #22 Feb 18 2022 02:43:45
%S A247903 1,5,13,29,53,93,149,237,357,541,789,1165,1669,2429,3445,4973,7013,
%T A247903 10077,14165,20301,28485,40765,57141,81709,114469,163613,229141,
%U A247903 327437,458501,655101,917237,1310445,1834725,2621149,3669717,5242573,7339717,10485437
%N A247903 Start with a single square; at n-th generation add a square at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)
%C A247903 Refer to A247618, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e."vertex to side" expansion version. Let us assign the label "1" to the square at the origin; at n-th generation add a square at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps 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. a(n) is the sum of all label values at n-th generation. The squares count is A001844. See illustration ("vertex to side" is equal to "side to vertex"). For n >= 1, (a(n) - a(n-1))/4 is A027383.
%H A247903 G. C. Greubel, <a href="/A247903/b247903.txt">Table of n, a(n) for n = 0..1000</a>
%H A247903 Kival Ngaokrajang, <a href="/A247903/a247903_1.pdf">Illustration of initial terms (vertex to side)</a>
%H A247903 Kival Ngaokrajang, <a href="/A247903/a247903.pdf">Illustration of initial terms (side to vertex)</a>
%H A247903 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,2).
%F A247903 a(0) = 1, for n >= 1, a(n) = 4*A027383(n) + a(n-1).
%F A247903 a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - _Colin Barker_, Sep 26 2014
%F A247903 G.f.: (1+3*x+2*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - _Colin Barker_, Sep 26 2014
%F A247903 A(n) = 2^(n/2+1)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) - (8*n + 27). - _G. C. Greubel_, Feb 18 2022
%t A247903 LinearRecurrence[{2,1,-4,2}, {1,5,13,29}, 51] (* _G. C. Greubel_, Feb 18 2022 *)
%o A247903 (PARI)
%o A247903 {
%o A247903 b=0; a=1; print1(1, ", ");
%o A247903 for (n=0, 50,
%o A247903      b=b+2^floor(n/2);
%o A247903      a=a+4*b;
%o A247903      print1(a, ", ")
%o A247903     )
%o A247903 }
%o A247903 (PARI)
%o A247903 Vec(-(2*x^3+2*x^2+3*x+1) / ((x-1)^2*(2*x^2-1)) + O(x^100)) \\ _Colin Barker_, Sep 26 2014
%o A247903 (Magma) [2^(n/2+1)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(8*n+27): n in [0..50]]; // _G. C. Greubel_, Feb 18 2022
%o A247903 (Sage) [2*2^(n/2)*((7+5*sqrt(2)) +(-1)^n*(7-5*sqrt(2))) -(8*n+27) for n in (0..50)] # _G. C. Greubel_, Feb 18 2022
%Y A247903 Vertex to vertex version: A061777, A247618, A247619, A247620.
%Y A247903 Vertex to side version: A101946, A247904, A247905.
%Y A247903 Cf. A001844, A027383.
%K A247903 nonn,easy
%O A247903 0,2
%A A247903 _Kival Ngaokrajang_, Sep 26 2014
%E A247903 More terms from _Colin Barker_, Sep 26 2014