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A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.

%I A343949 #10 May 11 2021 05:58:25
%S A343949 1,2,4,8,12,18,26,36,52,70,102,136,200,266,394,524,780,1038,1550,2064,
%T A343949 3088,4114,6162,8212,12308,16406,24598,32792,49176,65562,98330,131100,
%U A343949 196636,262174,393246,524320,786464,1048610,1572898,2097188,3145764,4194342,6291494
%N A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.
%C A343949 Expansion level n is the first 2^n segments of the curve, and can be taken as a graph with visited points as vertices and segments as edges.
%H A343949 Kevin Ryde, <a href="/A343949/b343949.txt">Table of n, a(n) for n = 0..1000</a>
%H A343949 Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index "Diameter".
%H A343949 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-2,2).
%F A343949 a(0) = 1.
%F A343949 a(2*n)   = 3*2^n + 2*n - 4 = 2*A275970(n-1), for n>=1.
%F A343949 a(2*n+1) = 4*2^n + 2*n - 2 = 2*A083706(n).
%F A343949 a(n+1) - a(n) = 2*A228693(n), for n>=1.
%F A343949 a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6.
%F A343949 G.f.: (1 + x - x^2 + x^3 - 4*x^5) / ((1+x) * (1-x)^2 * (1-2*x^2)).
%F A343949 G.f.: 2 - (1/2)/(1+x) - (9/2)/(1-x) + 1/(1-x)^2  + (3 + 4*x)/(1 - 2*x^2).
%e A343949 Curve n=4:
%e A343949      *--*  *--*
%e A343949      |  |  |  |        Start S to end E along segments.
%e A343949      *--*--*  *--*     Distance a(4) = 12,
%e A343949         |        |     which is also graph diameter.
%e A343949   E  *--*     S--*
%e A343949   |  |
%e A343949   *--*
%o A343949 (PARI) a(n) = if(n==0,1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);
%Y A343949 Cf. A275970, A083706, A228693.
%Y A343949 Cf. A332383, A332384 (curve coordinates).
%K A343949 nonn,easy
%O A343949 0,2
%A A343949 _Kevin Ryde_, May 05 2021