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A247904 Start with a single pentagon; at n-th generation add a pentagon 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 A247904 #18 Feb 18 2022 02:44:22
%S A247904 1,6,21,56,131,286,601,1236,2511,5066,10181,20416,40891,81846,163761,
%T A247904 327596,655271,1310626,2621341,5242776,10485651,20971406,41942921,
%U A247904 83885956,167772031,335544186,671088501,1342177136,2684354411,5368708966,10737418081
%N A247904 Start with a single pentagon; at n-th generation add a pentagon 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 A247904 Refer to A247619, 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 pentagon at the origin; at n-th generation add a pentagon 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 pentagons 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 pentagons count is A005891. See illustration. For n >= 1, (a(n) - a(n-1))/5 is A000225.
%H A247904 G. C. Greubel, <a href="/A247904/b247904.txt">Table of n, a(n) for n = 0..1000</a>
%H A247904 Kival Ngaokrajang, <a href="/A247904/a247904.pdf">Illustration of initial terms</a>
%H A247904 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A247904 a(0) = 1, for n >= 1, a(n) = 5*A000225(n) + a(n-1).
%F A247904 a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - _Colin Barker_, Sep 26 2014
%F A247904 G.f.: (1+2*x+2*x^2)/((1-x)^2*(1-2*x)). - _Colin Barker_, Sep 26 2014
%F A247904 From _G. C. Greubel_, Feb 18 2022: (Start)
%F A247904 a(n) = 10*2^n - (5*n + 9).
%F A247904 E.g.f.: 10*exp(2*x) - (9 + 5*x)*exp(x). (End)
%t A247904 LinearRecurrence[{4,-5,2}, {1,6,21}, 51] (* _G. C. Greubel_, Feb 18 2022 *)
%o A247904 (PARI)
%o A247904 a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))
%o A247904 for (n=0, 50, print1(a(n), ", "))
%o A247904 (PARI)
%o A247904 Vec(-(2*x^2+2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 26 2014
%o A247904 (Magma) [10*2^n -(5*n+9): n in [0..50]]; // _G. C. Greubel_, Feb 18 2022
%o A247904 (Sage) [5*2^(n+1) -(5*n+9) for n in (0..50)] # _G. C. Greubel_, Feb 18 2022
%Y A247904 Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.
%Y A247904 Cf. Vertex to side version: A101946, A247903, A247905.
%Y A247904 Cf. A000225, A005891.
%K A247904 nonn,easy
%O A247904 0,2
%A A247904 _Kival Ngaokrajang_, Sep 26 2014
%E A247904 More terms from _Colin Barker_, Sep 26 2014