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%I #28 Apr 17 2018 03:48:23
%S 1,2,3,5,7,9,11,13,17,21,23,25,27,31,35,39,43,47,55,63,65,67,69,73,77,
%T 81,85,89,97,105,109,113,117,125,133,141,149,157,173,189,191,193,195,
%U 199,203,207,211,215,223,231,235,239,243,251,259,267,275,283,299,315,319,323,327,335,343,351,359,367,383,399,407,415,423,439,455,471,487,503,535,567
%N Toothpick sequence starting at the vertex of y=3*abs(x).
%C Consider the graph y=3*abs(x). The first toothpick extends vertically from (0,0) to (0,2). Each toothpick is of length 2 and is laid either horizontally or vertically.
%C Subsequent toothpicks are placed in a similar rule as A139250. Place toothpicks by the following rules:
%C - Toothpicks must always stay inside the graph of y=3*abs(x).
%C - Call the end of a toothpick exposed if it does not touch another toothpick, or the line y=3*abs(x)
%C - Each horizontal toothpick has its midpoint touching an exposed vertical toothpick
%C - If no horizontal toothpick can be laid, then a vertical toothpick should be laid on any exposed ends, from its end.
%C The sequence is the number of toothpicks laid after n rounds.
%C The structure is essentially the same as the Sierpinski's triangle but here every equilateral triangle is replaced with an isosceles triangle and then every isosceles triangle is replaced with seven toothpicks. There are infinitely many sequences of this type. - _Omar E. Pol_, Mar 12 2016
%H Christopher J. Shore, <a href="http://tube.geogebra.org/student/mr4Tpnl0A">Geogebra image of the toothpick sequence</a>.
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e The pattern is the total number of toothpicks laid after n rounds.
%e Following the rules above, the first round has 1 toothpick, the second and third rounds also have 1 toothpick, but the fourth and fifth round both have 2 toothpicks. Finding the total toothpicks placed in this pattern (1,1,1,2,2) gives 1,2,3,5,7. Subsequent rounds have this same pattern repeated from the emerging branches thus:
%e (1,1,1,2,2) ---> 1,2,3,5,7
%e 2*(1,1,1,2,2) ---> 9,11,13,17,21
%e 2*((1,1,1,2,2),2*(1,1,1,2,2)) ---> 23,25,27,31,35,39,43,47,55,63
%e 2*((1,1,1,2,2),2*(1,1,1,2,2),2*((1,1,1,2,2),2*(1,1,1,2,2))) ---> 65,67,69,73,77,81,85,89,97,105,109,113,117,125,133,141,149,157,173,189
%e Summation of 1*the sequence 1,1,1,2,2
%e (1)=1
%e 1+(1)=2
%e 2+(1)=3
%e 3+(2)=5
%e 5+(2)=7
%e Summation of 2*the sequence 1,1,1,2,2
%e 7+2(1)=9
%e 9+2(1)=11
%e 11+2(1)=13
%e 13+2(2)=17
%e 17+2(2)=21
%e Summation of 3*the sequence 1,1,1,2,2
%e 21+2(1)=23
%e 23+2(1)=25
%e 25+2(1)=27
%e 27+2(2)=31
%e 31+2(2)=35
%Y Cf. A047999, A151566, A139250.
%K nonn
%O 1,2
%A _Christopher J. Shore_, Dec 07 2015
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