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%I #18 Jul 31 2023 18:29:00
%S 2,3,2,5,3,2,7,5,8,3,2,9,7,12,5,13,8,11,3,2,11,9,16,7,19,12,17,5,18,
%T 13,21,8,19,11,14,3,2,13,11,20,9,25,16,23,7,26,19,31,12,29,17,22,5,23,
%U 18,31,13,34,21,29,8,27,19,30
%N Eisenstein array Ei(2,3).
%C In Eisenstein's notation this is the array for m=2 and n=3; see pp. 41-2 of the Eisenstein reference given for A064881. This is identical with the array for m=3, n=2, given in A064885, read backwards. The array for m=n=1 is A049456.
%C For n >= 1, the number of entries of row n >= 1 is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 5*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, also for the Wilf link) with root 2/3. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.
%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)=2, a(1, 1)=3.
%e {2,3}; {2,5,3}; {2,7,5,8,3}; {2,9,7,12,5,13,8,11,3}; ...
%e This binary subtree of rationals is built from 2/3; 2/5,5/3; 2/7,7/5,5/8,8/3; ...
%t a[1, 0] = 2; a[1, 1] = 3; a[n_ /; n >= 1, m_ /; m >= 0] := If[EvenQ[m], a[n, m] = a[n-1, m/2], a[n, m] = a[n-1, (m-1)/2] + a[n-1, (m+1)/2]]; Table[a[n, m], {n, 1, 6}, {m, 0, 2^(n-1)}] // Flatten (* _Jean-François Alcover_, Feb 27 2018 *)
%K nonn,easy,tabf
%O 1,1
%A _Wolfdieter Lang_, Oct 19 2001
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