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Start with a single hexagon; at n-th generation add a hexagon 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 #26 Aug 22 2024 15:37:38

%S 1,7,25,67,157,343,721,1483,3013,6079,12217,24499,49069,98215,196513,

%T 393115,786325,1572751,3145609,6291331,12582781,25165687,50331505,

%U 100663147,201326437,402653023,805306201,1610612563,3221225293,6442450759,12884901697

%N Start with a single hexagon; at n-th generation add a hexagon 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 hexagon; at n-th generation add a hexagon at each expandable vertex, i.e. a vertex such that the new added generations will not overlap to the existing ones, but overlapping among new generations are allowed. The non-overlapping squares will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. The hexagons count is A003215. See llustration. For n >= 1, (a(n) - a(n-1))/6 is A000225

%H Paolo Xausa, <a href="/A247620/b247620.txt">Table of n, a(n) for n = 0..1000</a>

%H Kival Ngaokrajang, <a href="/A247620/a247620_1.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) = 6*A000225(n) + a(n-1).

%F From _Colin Barker_, Sep 21 2014: (Start)

%F a(n) = (-11+3*2^(2+n)-6*n).

%F a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3).

%F G.f.: -(x+1)*(2*x+1) / ((x-1)^2*(2*x-1)).

%F (End)

%t A247620[n_] := 3*2^(n+2) - 6*n - 11; Array[A247620, 50, 0] (* or *)

%t LinearRecurrence[{4, -5, 2}, {1, 7, 25}, 50] (* _Paolo Xausa_, Aug 22 2024 *)

%o (PARI) a(n) = if (n<1,1,6*(2^n-1)+a(n-1))

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

%o (PARI) Vec(-(x+1)*(2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 21 2014

%Y Cf. A000225, A061777, A003215, A247618, A247619.

%K nonn,easy

%O 0,2

%A _Kival Ngaokrajang_, Sep 21 2014

%E More terms from _Colin Barker_, Sep 21 2014