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

Author

Omar E. Pol, Jan 09 2010

Keywords

• nonn

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

Examples

			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)
		

Links

• 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

Crossrefs

See A170891 for the first differences.

Cf. A139250, A170886, A170888, A170892.

Programs

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

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