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A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212. 0
0, 1, 1, 2, 2, 4, 5, 4, 2, 4, 6, 6, 8, 14, 15, 8, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 39, 16, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 95 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

It appears that both a(2) and a(2^k - 1) are odd numbers, for k >= 2. Other terms are even numbers.

LINKS

Table of n, a(n) for n=1..63.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

FORMULA

a(1)=0. a(n) = A187213(n)/2, for n >= 2.

It appears that a(2^k - 1) = A099035(k-1), for k >= 2.

EXAMPLE

At stage 1 we start in the first quadrant from a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1). There are no gulls in the structure, so a(1) = 0.

At stage 2 we place a gull (or G-toothpick) with its midpoint at (1,1) and its endpoints at (2,0) and (2,2), so a(2) = 1. There is only one exposed midpoint at (2,2).

At stage 3 we place a gull with its midpoint at (2,2), so a(3) = 1. There are two exposed endpoints.

At stage 4 we place two gulls, so a(4) = 2. There are two exposed endpoints.

At stage 5 we place two gulls, so a(5) = 2. There are four exposed endpoints.

And so on.

If written as a triangle begins:

0,

1,

1,2,

2,4,5,4,

2,4,6,6,8,14,15,8,

2,4,6,6,8,14,16,10,8,14,18,20,30,44,39,16,

2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,42,28,...

It appears that rows converge to A151688.

MAPLE

read("transforms3") ;

L := BFILETOLIST("b187212.txt") ;

A187213 := DIFF(L) ;

seq( op(n, A187213)/2, n=2..nops(A187213)) ; # R. J. Mathar, Mar 30 2011

CROSSREFS

Cf. A099035, A151550, A151688, A187210, A187211, A187212, A187213, A187220.

Sequence in context: A186053 A133082 A130265 * A179821 A292272 A292248

Adjacent sequences:  A187211 A187212 A187213 * A187215 A187216 A187217

KEYWORD

nonn,changed

AUTHOR

Omar E. Pol, Mar 22 2011, Apr 06 2011

STATUS

approved

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Last modified February 25 19:39 EST 2021. Contains 341618 sequences. (Running on oeis4.)