OFFSET
1,2
COMMENTS
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.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Illustration of the construction after 7 steps
Index entries for linear recurrences with constant coefficients, signature (3,-2).
FORMULA
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.
From Colin Barker, Nov 02 2018: (Start)
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)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rémy Sigrist, Nov 01 2018
STATUS
approved