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A321257 Start with an equilateral triangle, and repeatedly append along the triangles of the previous step equilateral triangles with half their side length that do not overlap with any prior triangle; a(n) gives the number of triangles appended at n-th step.

%I A321257 #18 Sep 08 2022 08:46:23
%S A321257 1,6,21,60,147,330,705,1464,2991,6054,12189,24468,49035,98178,196473,
%T A321257 393072,786279,1572702,3145557,6291276,12582723,25165626,50331441,
%U A321257 100663080,201326367,402652950,805306125,1610612484,3221225211,6442450674,12884901609,25769803488
%N A321257 Start with an equilateral triangle, and repeatedly append along the triangles of the previous step equilateral triangles with half their side length that do not overlap with any prior triangle; a(n) gives the number of triangles appended at n-th step.
%C A321257 The following diagram depicts the first three steps of the construction:
%C A321257               -     -     -     -
%C A321257              / \   / \   / \   / \
%C A321257             / 3 \ / 3 \ / 3 \ / 3 \
%C A321257            -------------------------
%C A321257           / \         / \         / \
%C A321257          / 3 \   2   /   \   2   / 3 \
%C A321257         -------     /     \     -------
%C A321257        / \   / \   /       \   / \   / \
%C A321257       / 3 \ / 3 \ /         \ / 3 \ / 3 \
%C A321257      -------------           -------------
%C A321257     / \         /             \         / \
%C A321257    / 3 \   2   /       1       \   2   / 3 \
%C A321257   -------     /                 \     -------
%C A321257        / \   /                   \   / \
%C A321257       / 3 \ /                     \ / 3 \
%C A321257      -------------------------------------
%C A321257           / \         / \         / \
%C A321257          / 3 \   2   / 3 \   2   / 3 \
%C A321257         -------     -------     -------
%C A321257              / \   / \   / \   / \
%C A321257             / 3 \ / 3 \ / 3 \ / 3 \
%C A321257            -------------------------
%C A321257 A triangle of step n+1 touches one or two triangles of step n.
%C A321257 The construction presents holes from the 3rd step onwards; these will be gradually filled in the subsequent steps.
%C A321257 The limiting construction is a hexagon; its area is 6 times the area of the initial triangle.
%C A321257 See A321237 for a similar sequence.
%H A321257 Rémy Sigrist, <a href="/A321257/a321257.png">Illustration of the construction after 7 steps</a>
%H A321257 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A321257 a(n) = 3*(2^(n-1) + 3*(2^(n-1)-n)) for any n > 1.
%F A321257 Sum_{n > 0} a(n) / 4^(n-1) = 6.
%F A321257 G.f.: x*(1 + 2*x + 2*x^2 + 4*x^3)/((1-2*x)*(1-x)^2). - _Vincenzo Librandi_, Nov 02 2018
%F A321257 a(n) - 4*a(n-1) + 5*a(n-2) - 2*a(n-3) = 0, with n>1. - _Vincenzo Librandi_, Nov 02 2018
%p A321257 1,seq(3*(2^(n-1)+3*(2^(n-1)-n)),n=2..35); # _Muniru A Asiru_, Nov 02 2018
%t A321257 CoefficientList[Series[(1 + 2 x + 2 x^2 + 4 x^3) / ((1 - 2 x) (1 - x)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 02 2018 *)
%o A321257 (PARI) a(n) = if (n==1, 1, 3*(2^(n-1) + 3*(2^(n-1)-n)))
%o A321257 (Magma) [1] cat [3*(2^(n-1) + 3*(2^(n-1)-n)): n in [2..35]]; // _Vincenzo Librandi_, Nov 02 2018
%o A321257 (GAP) Concatenation([1],List([2..35],n->3*(2^(n-1)+3*(2^(n-1)-n)))); # _Muniru A Asiru_, Nov 02 2018
%Y A321257 Cf. A321237.
%K A321257 nonn
%O A321257 1,2
%A A321257 _Rémy Sigrist_, Nov 01 2018