

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 nth step.


2



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

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^(n1) + 3 * (2^(n2)1)) for any n > 1.
a(n) = 4 * A154117(n1) for any n > 1.
Sum_{n > 0} a(n) / 4^(n1) = 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(n1)  2*a(n2) for n>3.
(End)


PROG

(PARI) a(n) = if (n==1, return (1), return (4*( 2^(n1) + 3 * floor( (2^(n2)1) ) )))
(PARI) Vec(x*(1 + 2*x)*(1 + 3*x) / ((1  x)*(1  2*x)) + O(x^40)) \\ Colin Barker, Nov 02 2018


CROSSREFS

Cf. A154117, A321257.
Sequence in context: A350144 A028553 A100182 * A328535 A358247 A119515
Adjacent sequences: A321234 A321235 A321236 * A321238 A321239 A321240


KEYWORD

nonn,easy


AUTHOR

Rémy Sigrist, Nov 01 2018


STATUS

approved



