login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A153000 Toothpick sequence in the first quadrant. 31
0, 1, 2, 3, 5, 8, 10, 11, 13, 16, 19, 23, 30, 38, 42, 43, 45, 48, 51, 55, 62, 70, 75, 79, 86, 95, 105, 120, 142, 162, 170, 171, 173, 176, 179, 183, 190, 198, 203, 207, 214, 223, 233, 248, 270, 290, 299, 303, 310, 319, 329, 344, 366, 387 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Omar E. Pol, Nov 29 2009: (Start)
At stage 0, we start from a horizontal half toothpick at [(0,1),(1,1)]. This half toothpick represents one of the two components of the second toothpick placed in the toothpick structure of A139250. Consider only the toothpicks of length 2, so a(0) = 0.
At stage 1 we place an orthogonal toothpick of length 2 centered at the end, so a(1) = 1.
In each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
The sequence gives the number of toothpicks after n stages. Note that this sequence contains even numbers and odd numbers, the same as A152978 (the first differences) which gives the number of toothpicks added at n-th stage. For more information see A139250. (End)
A079559 gives the parity of this sequence, if n >= 1. - Omar E. Pol, Aug 13 2013
REFERENCES
D. 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
LINKS
David Applegate, The movie version
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.], which is also available at arXiv:1004.3036v2, [math.CO], 2010.
Omar E. Pol, Illustration of initial terms [From Omar E. Pol, Nov 29 2009]
FORMULA
a(n) = (A139250(n+2)-3)/4 = (A152998(n+1)-1)/2.
G.f.: (1+x)*(Product_{k>=1} (1+x^(2^k-1)+2*x^(2^k))-1)/((1-x)*(1+2*x)). - N. J. A. Sloane, May 20 2009
Contribution from Omar E. Pol, Oct 01 2011: (Start)
a(n) = A152998(n+1) + A153003(n+1) - A139250(n+2) + 1.
a(n) = A139250(n+2) - A153003(n+1) - 2.
a(n) = A153003(n+1) - A152998(n+1).
(End)
a(n) = (A187220(n+3) - 7)/8. - Omar E. Pol, Feb 16 2013
MAPLE
G := (1+x)*(mul(1+x^(2^k-1)+2*x^(2^k), k=1..20)-1)/((1-x)*(1+2*x)); # N. J. A. Sloane, May 20 2009
PROG
(Python)
def msb(n):
t=0
while n>>t>0: t+=1
return 2**(t - 1)
def a139250(n):
k=(2*msb(n)**2 + 1)//3
return 0 if n==0 else k if n==msb(n) else k + 2*a139250(n - msb(n)) + a139250(n - msb(n) + 1) - 1
def a(n): return 0 if n==0 else (a139250(n + 2) - 3)//4
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 01 2017
CROSSREFS
Cf. A152998, A160406. - Omar E. Pol, Nov 29 2009
Sequence in context: A189143 A047605 A295085 * A222172 A326379 A099107
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 16 2008, Dec 20 2008, Jan 02 2009
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 21 04:30 EST 2024. Contains 370219 sequences. (Running on oeis4.)