A153000 Toothpick sequence in the first quadrant.

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

Offset: 0

Author

Omar E. Pol, Dec 16 2008, Dec 20 2008, Jan 02 2009

Keywords

• nonn

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]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A139250, A139251, A152978, A153006.

Cf. A152998, A160406. - Omar E. Pol, Nov 29 2009

Programs

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

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

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