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A247619 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex; a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

%I A247619 #34 Aug 21 2024 12:27:03
%S A247619 1,6,16,36,66,116,186,296,446,676,986,1456,2086,3036,4306,6216,8766,
%T A247619 12596,17706,25376,35606,50956,71426,102136,143086,204516,286426,
%U A247619 409296,573126,818876,1146546,1638056,2293406,3276436,4587146,6553216,9174646,13106796
%N A247619 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex; a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)
%C A247619 Inspired by A061777, let us assign the label "1" to an origin pentagon; at the n-th generation add a pentagon at each expandable vertex, i.e., a vertex such that the new added generations will not overlap existing ones, but overlapping among new generations is allowed. Each nonoverlapping pentagon will have the same label value as its predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. The pentagon count is A005891. See illustration. [Edited for grammar/style by _Peter Munn_, Jan 14 2023]
%H A247619 Paolo Xausa, <a href="/A247619/b247619.txt">Table of n, a(n) for n = 0..1000</a>
%H A247619 Kival Ngaokrajang, <a href="/A247619/a247619_1.pdf">Illustration of initial terms</a>
%H A247619 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,2).
%F A247619 a(0) = 1, for n >= 1, a(n) = 5*A027383(n-1) + a(n-1). [Offset corrected by _Peter Munn_, Apr 20 2023]
%F A247619 a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+2*a(n-4). G.f.: -(2*x^3+3*x^2+4*x+1) / ((x-1)^2*(2*x^2-1)). - _Colin Barker_, Sep 21 2014
%t A247619 LinearRecurrence[{2, 1, -4, 2}, {1, 6, 16, 36}, 50] (* _Paolo Xausa_, Aug 21 2024 *)
%o A247619 (PARI)
%o A247619 {
%o A247619 b=0;a=1;print1(1,", ");
%o A247619 for (n=0,50,
%o A247619      b=b+2^floor(n/2);
%o A247619      a=a+5*b;
%o A247619      print1(a,", ")
%o A247619     )
%o A247619 }
%o A247619 (PARI)
%o A247619 Vec(-(2*x^3+3*x^2+4*x+1)/((x-1)^2*(2*x^2-1)) + O(x^100)) \\ _Colin Barker_, Sep 21 2014
%Y A247619 Cf. A005891, A027383, A061777, A247618, A247620.
%Y A247619 See A358632 for a related concept.
%K A247619 nonn,easy
%O A247619 0,2
%A A247619 _Kival Ngaokrajang_, Sep 21 2014
%E A247619 More terms from _Colin Barker_, Sep 21 2014