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A247905 Start with a single hexagon; at n-th generation add a hexagon 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 A247905 #17 Feb 18 2022 02:43:36
%S A247905 1,7,19,43,79,139,223,355,535,811,1183,1747,2503,3643,5167,7459,10519,
%T A247905 15115,21247,30451,42727,61147,85711,122563,171703,245419,343711,
%U A247905 491155,687751,982651,1375855,1965667,2752087,3931723,5504575,7863859,11009575,15728155,22019599,31456771,44039671,62914027,88079839,125828563,176160199,251657659
%N A247905 Start with a single hexagon; at n-th generation add a hexagon 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 A247905 Refer to A247620, 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" the hexagon at the origin; at n-th generation add a hexagon 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 hexagons 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 hexagons count is A003215. See illustration. For n >= 1, (a(n) - a(n-1))/6 is A027383.
%H A247905 G. C. Greubel, <a href="/A247905/b247905.txt">Table of n, a(n) for n = 0..1000</a>
%H A247905 Kival Ngaokrajang, <a href="/A247905/a247905.pdf">Illustration of initial terms</a>
%H A247905 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,2).
%F A247905 a(0) = 1, for n >= 1, a(n) = 6*A027383(n) + a(n-1).
%F A247905 a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - _Colin Barker_, Sep 26 2014
%F A247905 G.f.: (1+5*x+4*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - _Colin Barker_, Sep 26 2014
%F A247905 a(n) = 3*2^(n/2)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) -12*n - 41. - _G. C. Greubel_, Feb 18 2022
%t A247905 LinearRecurrence[{2,1,-4,2}, {1,7,19,43}, 50] (* _G. C. Greubel_, Feb 17 2022 *)
%o A247905 (PARI)
%o A247905 {
%o A247905 b=0; a=1; print1(1, ", ");
%o A247905 for (n=0, 50,
%o A247905      b=b+2^floor(n/2);
%o A247905      a=a+6*b;
%o A247905      print1(a, ", ")
%o A247905     )
%o A247905 }
%o A247905 (PARI)
%o A247905 Vec(-(2*x^3+4*x^2+5*x+1)/((x-1)^2*(2*x^2-1)) + O(x^100)) \\ _Colin Barker_, Sep 26 2014
%o A247905 (Magma) [3*2^(n/2)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(12*n+41): n in [0..50]]; // _G. C. Greubel_, Feb 17 2022
%o A247905 (Sage) [3*2^(n/2)*((7+5*sqrt(2)) + (-1)^n*(7-5*sqrt(2))) -(12*n+41) for n in (0..50)] # _G. C. Greubel_, Feb 17 2022
%Y A247905 Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.
%Y A247905 Cf. Vertex to side version: A101946, A247903, A247904.
%Y A247905 Cf. A003215, A027383.
%K A247905 nonn,easy
%O A247905 0,2
%A A247905 _Kival Ngaokrajang_, Sep 26 2014