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A321237
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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.
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2
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1, 8, 28, 68, 148, 308, 628, 1268, 2548, 5108, 10228, 20468, 40948, 81908, 163828, 327668, 655348, 1310708, 2621428, 5242868, 10485748, 20971508, 41943028, 83886068, 167772148, 335544308, 671088628, 1342177268, 2684354548, 5368709108, 10737418228, 21474836468
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OFFSET
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1,2
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COMMENTS
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The following diagram depicts the first three steps of the construction:
+----+----+----+----+
| 3 | 3 | 3 | 3 |
+----+----+----+----+----+----+
| 3 | | | 3 |
+----+----+ 2 | 2 +----+----+
| 3 | 3 | | | 3 | 3 |
+----+----+----+---------+---------+----+----+----+
| 3 | | | | 3 |
+----+ 2 | | 2 +----+
| 3 | | | | 3 |
+----+---------+ 1 +---------+----+
| 3 | | | | 3 |
+----+ 2 | | 2 +----+
| 3 | | | | 3 |
+----+----+----+---------+---------+----+----+----+
| 3 | 3 | | | 3 | 3 |
+----+----+ 2 | 2 +----+----+
| 3 | | | 3 |
+----+----+----+----+----+----+
| 3 | 3 | 3 | 3 |
+----+----+----+----+
A square of step n+1 touches one or two squares of step n.
The limiting construction is an octagon (truncated square); its area is 7 times the area of the initial square.
See A321257 for a similar sequence.
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LINKS
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FORMULA
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a(n) = 4 * (2^(n-1) + 3 * (2^(n-2)-1)) for any n > 1.
a(n) = 4 * A154117(n-1) for any n > 1.
Sum_{n > 0} a(n) / 4^(n-1) = 7.
G.f.: x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)).
a(n) = 5*2^n - 12 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
(End)
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PROG
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(PARI) a(n) = if (n==1, return (1), return (4*( 2^(n-1) + 3 * floor( (2^(n-2)-1) ) )))
(PARI) Vec(x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 02 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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