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%I #41 Jul 31 2023 18:30:27
%S 1,2,1,3,2,1,4,3,5,2,1,5,4,7,3,8,5,7,2,1,6,5,9,4,11,7,10,3,11,8,13,5,
%T 12,7,9,2,1,7,6,11,5,14,9,13,4,15,11,18,7,17,10,13,3,14,11,19,8,21,13,
%U 18,5,17,12,19,7,16,9,11,2
%N Eisenstein array Ei(1,2).
%C In Eisenstein's notation this is the array for m=1 and n=2; see example in given reference p. 42. The array for m=n=1 is A049456.
%C For n >= 1, the number of entries of row n is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 3*A007051(n-1).
%C The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the subtree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the reference see A002487 and the link) with root 1/2. The composition rule for this tree is i/j -> i/(i+j), (i+j)/j.
%H R. Backhouse, J. F. Ferreira, <a href="https://doi.org/10.1016/j.scico.2010.05.006">On Euclid’s algorithm and elementary number theory</a>, Sci. Comput. Program. 76, No. 3, 160-180 (2011).
%H N. Calkin and H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/website/recounting.pdf">Recounting the Rationals</a>, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
%H F. G. M. Eisenstein, <a href="http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=08-verh/1850&seite:int=36">Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden</a>, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850) 36-42, Feb 18, 1850. Werke, II, pp. 705-711.
%H <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F a(n, m) = a(n-1, m/2) if m is even, else a(n, m) = a(n-1, (m-1)/2) + a(n-1, (m+1)/2), a(1, 0)=1, a(1, 1)=2.
%e {1,2};
%e {1,3,2};
%e {1,4,3,5,2};
%e {1,5,4,7,3,8,5,7,2}; ...
%e This binary subtree of rationals is built from
%e 1/2;
%e 1/3, 3/2;
%e 1/4, 4/3, 3/5, 5/2; ...
%t nmax = 6; a[n_, m_?EvenQ] := a[n - 1, m/2]; a[n_, m_?OddQ] := a[n, m] = a[n - 1, (m - 1)/2] + a[n - 1, (m + 1)/2]; a[1, 0] = 1; a[1, 1] = 2; Flatten[ Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n - 1)}]] (* _Jean-François Alcover_, Sep 27 2011 *)
%t eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;
%t Flatten@NestList[eisen, {1, 2}, 6] (* _Harlan J. Brothers_, Feb 18 2015 *)
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Oct 19 2001
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