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%I #22 Feb 18 2022 02:43:45
%S 1,5,13,29,53,93,149,237,357,541,789,1165,1669,2429,3445,4973,7013,
%T 10077,14165,20301,28485,40765,57141,81709,114469,163613,229141,
%U 327437,458501,655101,917237,1310445,1834725,2621149,3669717,5242573,7339717,10485437
%N 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 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 G. C. Greubel, <a href="/A247903/b247903.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A247903/a247903_1.pdf">Illustration of initial terms (vertex to side)</a>
%H Kival Ngaokrajang, <a href="/A247903/a247903.pdf">Illustration of initial terms (side to vertex)</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,2).
%F a(0) = 1, for n >= 1, a(n) = 4*A027383(n) + a(n-1).
%F a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - _Colin Barker_, Sep 26 2014
%F G.f.: (1+3*x+2*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - _Colin Barker_, Sep 26 2014
%F 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 LinearRecurrence[{2,1,-4,2}, {1,5,13,29}, 51] (* _G. C. Greubel_, Feb 18 2022 *)
%o (PARI)
%o {
%o b=0; a=1; print1(1, ", ");
%o for (n=0, 50,
%o b=b+2^floor(n/2);
%o a=a+4*b;
%o print1(a, ", ")
%o )
%o }
%o (PARI)
%o 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 (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 (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 Vertex to vertex version: A061777, A247618, A247619, A247620.
%Y Vertex to side version: A101946, A247904, A247905.
%Y Cf. A001844, A027383.
%K nonn,easy
%O 0,2
%A _Kival Ngaokrajang_, Sep 26 2014
%E More terms from _Colin Barker_, Sep 26 2014
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