A170890 Toothpick sequence similar to A160406, but always staying outside the wedge, starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.

0, 1, 2, 4, 7, 10, 14, 21, 29, 37, 43, 53, 61, 71, 83, 103, 123, 139, 151, 165, 173, 183, 195, 215, 235, 253, 271, 295, 317, 345, 385, 441, 493, 531, 559, 581, 589, 599, 611, 631, 651, 669, 687, 711, 733, 761, 801, 857, 909, 949, 983, 1015, 1037, 1065, 1105, 1161

Offset

0,3

Comments

The initial half-tookpick makes an angle of 90 degrees w.r.t. the wedge's direction. This breaks the symmetry and explains the changing parity of the terms. - M. F. Hasler, Jan 29 2013

Links

Table of n, a(n) for n=0..55.

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

Example

From M. F. Hasler, Jan 29 2013: (Start)
The first steps are illustrated as follows, where two vertical "|" or three horizontal "_" correspond to one single full toothpick:
:                                ___ ___  |___ ___|
:                 ___    |___|    |___|   | |___| |
:   _      |_      |_    | |_|    | |_|   | | |_|
:   /\     |/\     |/\     |/\   ¯¯¯|/\   |¯¯¯|/\
:  /  \    /  \    /  \    /  \     /  \      /  \
:
: a(0)=0, a(1)=1, a(2)=2, a(3)=4, a(5)=7, a(6)=10, ... (End)

Prog

(PARI) A170890(n, print_all=0)={ my( cnt=n>0, ee=[[1, 1]], p=Set(vector(2*n-cnt, k, k-n-abs(k-n)*I)), c, d); for(i=2, n, print_all & print1(cnt", "); p=setunion(p, Set(Mat(ee~)[, 1])); my(ne=[]); for(k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); cnt+=#ee); cnt} \\ - M. F. Hasler, Jan 29 2013

Crossrefs

See A170891 for the first differences.

Cf. A139250, A170886, A170888, A170892.

Sequence in context: A094281 A076101 A288243 * A079963 A261252 A056750

Adjacent sequences: A170887 A170888 A170889 * A170891 A170892 A170893

Keyword

nonn

Author

Omar E. Pol, Jan 09 2010

Extensions

a(9) corrected by Omar E. Pol, following an observation by Kevin Ryde, Jan 29 2013

Terms beyond a(9) from M. F. Hasler, Jan 29 2013

Status

approved