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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.
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%I #10 May 11 2021 05:58:25

%S 1,2,4,8,12,18,26,36,52,70,102,136,200,266,394,524,780,1038,1550,2064,

%T 3088,4114,6162,8212,12308,16406,24598,32792,49176,65562,98330,131100,

%U 196636,262174,393246,524320,786464,1048610,1572898,2097188,3145764,4194342,6291494

%N 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 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 Kevin Ryde, <a href="/A343949/b343949.txt">Table of n, a(n) for n = 0..1000</a>

%H Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index "Diameter".

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-2,2).

%F a(0) = 1.

%F a(2*n) = 3*2^n + 2*n - 4 = 2*A275970(n-1), for n>=1.

%F a(2*n+1) = 4*2^n + 2*n - 2 = 2*A083706(n).

%F a(n+1) - a(n) = 2*A228693(n), for n>=1.

%F 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 G.f.: (1 + x - x^2 + x^3 - 4*x^5) / ((1+x) * (1-x)^2 * (1-2*x^2)).

%F G.f.: 2 - (1/2)/(1+x) - (9/2)/(1-x) + 1/(1-x)^2 + (3 + 4*x)/(1 - 2*x^2).

%e Curve n=4:

%e *--* *--*

%e | | | | Start S to end E along segments.

%e *--*--* *--* Distance a(4) = 12,

%e | | which is also graph diameter.

%e E *--* S--*

%e | |

%e *--*

%o (PARI) a(n) = if(n==0,1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);

%Y Cf. A275970, A083706, A228693.

%Y Cf. A332383, A332384 (curve coordinates).

%K nonn,easy

%O 0,2

%A _Kevin Ryde_, May 05 2021