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A265384 Toothpick sequence starting at the vertex of y=3*abs(x).

%I A265384 #34 Jun 07 2025 00:38:39
%S A265384 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 A265384 81,85,89,97,105,109,113,117,125,133,141,149,157,173,189,191,193,195,
%U A265384 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 A265384 Toothpick sequence starting at the vertex of y=3*abs(x).
%C A265384 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 A265384 Subsequent toothpicks are placed in a similar rule as A139250. Place toothpicks by the following rules:
%C A265384 - Toothpicks must always stay inside the graph of y=3*abs(x).
%C A265384 - Call the end of a toothpick exposed if it does not touch another toothpick, or the line y=3*abs(x)
%C A265384 - Each horizontal toothpick has its midpoint touching an exposed vertical toothpick
%C A265384 - If no horizontal toothpick can be laid, then a vertical toothpick should be laid on any exposed ends, from its end.
%C A265384 The sequence is the number of toothpicks laid after n rounds.
%C A265384 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 A265384 Christopher J. Shore, <a href="/A265384/a265384.jpg">Geogebra image of the toothpick sequence</a>.
%H A265384 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e A265384 The pattern is the total number of toothpicks laid after n rounds.
%e A265384 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 A265384 (1,1,1,2,2) ---> 1,2,3,5,7
%e A265384 2*(1,1,1,2,2) ---> 9,11,13,17,21
%e A265384 2*((1,1,1,2,2),2*(1,1,1,2,2)) ---> 23,25,27,31,35,39,43,47,55,63
%e A265384 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 A265384 Summation of 1*the sequence 1,1,1,2,2
%e A265384   (1)=1
%e A265384 1+(1)=2
%e A265384 2+(1)=3
%e A265384 3+(2)=5
%e A265384 5+(2)=7
%e A265384 Summation of 2*the sequence 1,1,1,2,2
%e A265384 7+2(1)=9
%e A265384 9+2(1)=11
%e A265384 11+2(1)=13
%e A265384 13+2(2)=17
%e A265384 17+2(2)=21
%e A265384 Summation of 3*the sequence 1,1,1,2,2
%e A265384 21+2(1)=23
%e A265384 23+2(1)=25
%e A265384 25+2(1)=27
%e A265384 27+2(2)=31
%e A265384 31+2(2)=35
%Y A265384 Cf. A047999, A151566, A139250.
%K A265384 nonn
%O A265384 1,2
%A A265384 _Christopher J. Shore_, Dec 07 2015