A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

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

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

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: A130265 A368156 A359045 * A179821 A292272 A292248

Adjacent sequences: A187211 A187212 A187213 * A187215 A187216 A187217

Keyword

nonn

Author

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

Status

approved