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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.
5
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
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.]
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.
Sequence in context: A094281 A076101 A288243 * A079963 A261252 A056750
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