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%I #35 Feb 24 2021 02:48:19
%S 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,
%T 22,28,32,42,66,66,34,4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,36,
%U 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
%N First differences of the toothpick sequence A170892.
%C 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
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%e From _Omar E. Pol_, Jan 30 2013 (Start):
%e 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.
%e 0;
%e 1;
%e 1;
%e 2;
%e 4,4;
%e 4,8,10,10;
%e 4,8,10,12,12,22,26,18;
%e 4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,34;
%e 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;
%e 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,...
%e (End)
%o (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
%Y Cf. A139250, A139251, A160407, A170887, A170889, A170891, A170892.
%K nonn,tabf
%O 0,4
%A _Omar E. Pol_, Jan 09 2010
%E Values beyond a(10) from _M. F. Hasler_, Jan 30 2013
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