A170893 First differences of the toothpick sequence A170892.

0, 1, 1, 2, 4, 4, 4, 8, 10, 10, 4, 8, 10, 12, 12, 22, 26, 18, 4, 8, 10, 12, 12, 22, 26, 20, 12, 22, 28, 32, 42, 66, 66, 34, 4, 8, 10, 12, 12, 22, 26, 20, 12, 22, 28, 32, 42, 66, 66, 36, 12, 22, 28, 32, 42, 66, 68, 48, 42, 68, 84, 102, 146, 194, 162, 66, 4, 8, 10, 12, 12, 22, 26, 20, 12, 22, 28, 32, 42, 66, 66, 36, 12, 22, 28, 32, 42, 66, 68, 48, 42, 68, 84

Offset: 0

Author

Omar E. Pol, Jan 09 2010

Keywords

• nonn
• tabf

Comments

This describes how many toothpicks are added at each step (as to form the upper bar of a T) at all "exposed" endpoints, starting from an initial configuration with a vertical toothpick whose lower endpoint rests on the top of the conic region { (x,y): y < -|x| } into which the toothpicks may not extend. - M. F. Hasler, Jan 30 2013

Examples

			From _Omar E. Pol_, Jan 30 2013 (Start):
If written as an irregular triangle in which rows 0..2 have length 1, it appears that row j has length 2^(j-3), if j >= 3.
0;
1;
1;
2;
4,4;
4,8,10,10;
4,8,10,12,12,22,26,18;
4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,34;
4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,36,12,22,28,32,42,66,68,48,42,68,84,102,146,194,162,66;
4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,36,12,22,28,32,42,66,68,48,42,68,84,...
(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

Cf. A139250, A139251, A160407, A170887, A170889, A170891, A170892.

Programs

PARI

A170893(n, print_all=0)={my( ee=[[2*I, I]], p=Set( concat( vector( 2*n-(n>0), k, k-n-abs(k-n)*I ), I ))); print_all & print1("1,1"); for(i=3, n, p=setunion(p, Set(Mat(ee~)[, 1])); my(c, d, 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))); print_all & print1(","#ee)); (n>0)*#ee} \\ M. F. Hasler, Jan 30 2013

Extensions

Values beyond a(10) from M. F. Hasler, Jan 30 2013