%I #18 Feb 18 2022 02:44:22

%S 1,6,21,56,131,286,601,1236,2511,5066,10181,20416,40891,81846,163761,

%T 327596,655271,1310626,2621341,5242776,10485651,20971406,41942921,

%U 83885956,167772031,335544186,671088501,1342177136,2684354411,5368708966,10737418081

%N 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 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 G. C. Greubel, <a href="/A247904/b247904.txt">Table of n, a(n) for n = 0..1000</a>

%H Kival Ngaokrajang, <a href="/A247904/a247904.pdf">Illustration of initial terms</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).

%F a(0) = 1, for n >= 1, a(n) = 5*A000225(n) + a(n-1).

%F a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - _Colin Barker_, Sep 26 2014

%F G.f.: (1+2*x+2*x^2)/((1-x)^2*(1-2*x)). - _Colin Barker_, Sep 26 2014

%F From _G. C. Greubel_, Feb 18 2022: (Start)

%F a(n) = 10*2^n - (5*n + 9).

%F E.g.f.: 10*exp(2*x) - (9 + 5*x)*exp(x). (End)

%t LinearRecurrence[{4,-5,2}, {1,6,21}, 51] (* _G. C. Greubel_, Feb 18 2022 *)

%o (PARI)

%o a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))

%o for (n=0, 50, print1(a(n), ", "))

%o (PARI)

%o Vec(-(2*x^2+2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 26 2014

%o (Magma) [10*2^n -(5*n+9): n in [0..50]]; // _G. C. Greubel_, Feb 18 2022

%o (Sage) [5*2^(n+1) -(5*n+9) for n in (0..50)] # _G. C. Greubel_, Feb 18 2022

%Y Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.

%Y Cf. Vertex to side version: A101946, A247903, A247905.

%Y Cf. A000225, A005891.

%K nonn,easy

%O 0,2

%A _Kival Ngaokrajang_, Sep 26 2014

%E More terms from _Colin Barker_, Sep 26 2014