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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.)
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%I #34 Aug 21 2024 12:27:03

%S 1,6,16,36,66,116,186,296,446,676,986,1456,2086,3036,4306,6216,8766,

%T 12596,17706,25376,35606,50956,71426,102136,143086,204516,286426,

%U 409296,573126,818876,1146546,1638056,2293406,3276436,4587146,6553216,9174646,13106796

%N 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 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 Paolo Xausa, <a href="/A247619/b247619.txt">Table of n, a(n) for n = 0..1000</a>

%H Kival Ngaokrajang, <a href="/A247619/a247619_1.pdf">Illustration of initial terms</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) = 5*A027383(n-1) + a(n-1). [Offset corrected by _Peter Munn_, Apr 20 2023]

%F 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 LinearRecurrence[{2, 1, -4, 2}, {1, 6, 16, 36}, 50] (* _Paolo Xausa_, Aug 21 2024 *)

%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+5*b;

%o print1(a,", ")

%o )

%o }

%o (PARI)

%o 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 Cf. A005891, A027383, A061777, A247618, A247620.

%Y See A358632 for a related concept.

%K nonn,easy

%O 0,2

%A _Kival Ngaokrajang_, Sep 21 2014

%E More terms from _Colin Barker_, Sep 21 2014