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A337293 a(n) is the squared distance to the origin of the n-th vertex on an acute angled Babylonian spiral.

%I A337293 #51 Jan 01 2025 23:31:21
%S A337293 0,1,1,1,2,2,5,1,8,8,5,17,1,26,0,29,5,25,25,18,34,5,50,1,49,17,18,52,
%T A337293 5,85,2,90,18,61,125,13,148,10,153,20,98,125,41,145,4,148,18,85,170,
%U A337293 18,225,148,202,173,61,197,41,226,10,229,25,117,170,5,208,80
%N A337293 a(n) is the squared distance to the origin of the n-th vertex on an acute angled Babylonian spiral.
%C A337293 An acute angled Babylonian spiral is constructed by starting with a zero vector and progressively concatenating the next longest vector with integral endpoints on a Cartesian grid.  (The squares of the lengths of these vectors are A001481.)  The direction of the new vector is chosen to maximize the change in direction from the previous vector. The Babylonian spiral (A256111) minimizes this angle.
%H A337293 John Bailey, <a href="/A337293/b337293.txt">Table of n, a(n) for n = 0..10000</a>
%H A337293 John Bailey, Illustrations of <a href="/A337293/a337293.png">10</a>, <a href="/A337293/a337293_1.png">100</a>, <a href="/A337293/a337293_2.png">1000</a>, <a href="/A337293/a337293_3.png">10000</a>, <a href="/A337293/a337293_4.png">100000</a>, <a href="/A337293/a337293_5.png">1500000</a>, <a href="/A337293/a337293_6.png">10000000</a>
%H A337293 John Bailey, <a href="/A337293/a337293.py.txt">Python code with plots</a>
%F A337293 a(n) = A337311(n)^2 + A337312(n)^2.
%e A337293 The coordinates of the first few points are (0,0), (0,1), (1,0), (-1,0), (1,1), (-1,-1), (-1,2).
%o A337293 (Python) # See Bailey link.
%Y A337293 x-coordinates given in A337311. y-coordinates given in A337312.
%Y A337293 Cf. A001481, A256111.
%K A337293 nonn
%O A337293 0,5
%A A337293 _John Bailey_, Aug 21 2020