

A187214


Number of gulls (or Gtoothpicks) added at nth 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
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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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)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(k1), for k >= 2.


EXAMPLE

At stage 1 we start in the first quadrant from a Qtoothpick 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 Gtoothpick) 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



