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A238549 a(n) is one fourth of the total number of free ends of 4 line segments expansion at n iterations (see Comments lines for definition). 2
1, 2, 6, 8, 16, 20, 36, 44, 76, 92, 156, 188, 316, 380, 636, 764, 1276, 1532, 2556, 3068, 5116, 6140, 10236, 12284, 20476, 24572, 40956, 49148, 81916, 98300, 163836, 196604, 327676, 393212, 655356, 786428, 1310716, 1572860, 2621436, 3145724, 5242876, 6291452, 10485756 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The initial pattern consists of 4 straight line segments which are the radii of a square. The next generations are scaled down by a factor of 1/sqrt(2) and rotated by an angle of Pi/4. Their free ends are the ends of elements that do not contact or cross the other ones. Overlaps among different generations are prohibited. See illustration in the links.

We take the official definition to be that provided by the PARI program. From this the assertions in the Formula section follow (they were formerly stated as conjectures). - N. J. A. Sloane, Feb 24 2019

From Georg Fischer, Feb 20 2019: (Start)

The following pattern can be seen for a(n) in base 2:

   n         a(n)

  ==  ==================

   1    1 =          1_2

   2    2 =         10_2

   3    6 =        110_2

   4    8 =       1000_2

   5   16 =      10000_2

   6   20 =      10100_2

   7   36 =     100100_2

   8   44 =     101100_2

   9   76 =    1001100_2

  10   92 =    1011100_2

  11  156 =   10011100_2

  12  188 =   10111100_2

  13  316 =  100111100_2

  14  380 =  101111100_2

  15  636 = 1001111100_2

  16  764 = 1011111100_2

(End)

LINKS

Table of n, a(n) for n=1..43.

Kival Ngaokrajang, Illustration of initial terms

FORMULA

a(n) =  1 + Sum_{i=1..n-1} A143095(i).

G.f.: x*(2*x^2+x+1) / ((x-1)*(2*x^2-1)). - Colin Barker, May 02 2015

From Georg Fischer, Feb 20 2019: (Start)

With p = floor((n + 2) / 2) for n >= 4: if n even then a(n) = 2^p + 4 * (2^(p - 4) - 1); if n odd then a(n) = 2^p + 4 * (2^(p - 3) - 1).

a(n) = a(n - 1) + 2 * a(n - 2) - 2 * a(n - 3).

(End)

EXAMPLE

The first numbers of free ends (4*a(n)) are 4, 8, 24, 32, 64, 80, 144, 176, 304, 368, 624, ...

PROG

(PARI) {print1(1, ", "); for (n=1, 100, s=1; for (i=0, n-1, s=s+(5-3*(-1)^i)*2^(1/4*(2*i-1+(-1)^i))/2); print1(s, ", "))}

(Sage)

def a():

    s, n = 2, 1

    yield 1

    while True:

        yield s

        s += (5-3*(-1)^n)*2^((2*n-1+(-1)^n)//4)//2

        n += 1

A238549 = a(); [next(A238549) for _ in range(43)] # Peter Luschny, Feb 24 2019

CROSSREFS

Cf. A143095, A256641.

Sequence in context: A326300 A266074 A191822 * A237502 A279726 A331974

Adjacent sequences:  A238546 A238547 A238548 * A238550 A238551 A238552

KEYWORD

nonn

AUTHOR

Kival Ngaokrajang, May 01 2015

STATUS

approved

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Last modified February 17 12:32 EST 2020. Contains 331996 sequences. (Running on oeis4.)