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A321237 Start with a square of dimension 1 X 1, and repeatedly append along the squares of the previous step squares with half their side length that do not overlap with any prior square; a(n) gives the number of squares appended at n-th step.

%I A321237 #17 Nov 02 2018 12:02:02
%S A321237 1,8,28,68,148,308,628,1268,2548,5108,10228,20468,40948,81908,163828,
%T A321237 327668,655348,1310708,2621428,5242868,10485748,20971508,41943028,
%U A321237 83886068,167772148,335544308,671088628,1342177268,2684354548,5368709108,10737418228,21474836468
%N A321237 Start with a square of dimension 1 X 1, and repeatedly append along the squares of the previous step squares with half their side length that do not overlap with any prior square; a(n) gives the number of squares appended at n-th step.
%C A321237 The following diagram depicts the first three steps of the construction:
%C A321237                        +----+----+----+----+
%C A321237                        |  3 |  3 |  3 |  3 |
%C A321237                   +----+----+----+----+----+----+
%C A321237                   |  3 |         |         |  3 |
%C A321237              +----+----+    2    |    2    +----+----+
%C A321237              |  3 |  3 |         |         |  3 |  3 |
%C A321237         +----+----+----+---------+---------+----+----+----+
%C A321237         |  3 |         |                   |         |  3 |
%C A321237         +----+    2    |                   |    2    +----+
%C A321237         |  3 |         |                   |         |  3 |
%C A321237         +----+---------+         1         +---------+----+
%C A321237         |  3 |         |                   |         |  3 |
%C A321237         +----+    2    |                   |    2    +----+
%C A321237         |  3 |         |                   |         |  3 |
%C A321237         +----+----+----+---------+---------+----+----+----+
%C A321237              |  3 |  3 |         |         |  3 |  3 |
%C A321237              +----+----+    2    |    2    +----+----+
%C A321237                   |  3 |         |         |  3 |
%C A321237                   +----+----+----+----+----+----+
%C A321237                        |  3 |  3 |  3 |  3 |
%C A321237                        +----+----+----+----+
%C A321237 A square of step n+1 touches one or two squares of step n.
%C A321237 The limiting construction is an octagon (truncated square); its area is 7 times the area of the initial square.
%C A321237 See A321257 for a similar sequence.
%H A321237 Colin Barker, <a href="/A321237/b321237.txt">Table of n, a(n) for n = 1..1000</a>
%H A321237 Rémy Sigrist, <a href="/A321237/a321237.png">Illustration of the construction after 7 steps</a>
%H A321237 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A321237 a(n) = 4 * (2^(n-1) + 3 * (2^(n-2)-1)) for any n > 1.
%F A321237 a(n) = 4 * A154117(n-1) for any n > 1.
%F A321237 Sum_{n > 0} a(n) / 4^(n-1) = 7.
%F A321237 From _Colin Barker_, Nov 02 2018: (Start)
%F A321237 G.f.: x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)).
%F A321237 a(n) = 5*2^n - 12 for n>1.
%F A321237 a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
%F A321237 (End)
%o A321237 (PARI) a(n) = if (n==1, return (1), return (4*( 2^(n-1) + 3 * floor( (2^(n-2)-1) ) )))
%o A321237 (PARI) Vec(x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, Nov 02 2018
%Y A321237 Cf. A154117, A321257.
%K A321237 nonn,easy
%O A321237 1,2
%A A321237 _Rémy Sigrist_, Nov 01 2018