%I #19 Jul 31 2023 18:30:08

%S 2,1,2,3,1,2,5,3,4,1,2,7,5,8,3,7,4,5,1,2,9,7,12,5,13,8,11,3,10,7,11,4,

%T 9,5,6,1,2,11,9,16,7,19,12,17,5,18,13,21,8,19,11,14,3,13,10,17,7,18,

%U 11,15,4,13,9,14,5,11,6,7,1

%N Eisenstein array Ei(2,1).

%C In Eisenstein's notation this is the array for m=2 and n=1; see pp. 41-2 of the Eisenstein reference given for A064881. This is identical with the array for m=1, n=2, given in A064881, read backwards. 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 (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 2/1. 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)=1.

%e {2,1}; {2,3,1}; {2,5,3,4,1}; {2,7,5,8,3,7,4,5,1}; ...

%e This binary subtree of rationals is built from 2/1; 2/3, 3/1; 2/5, 5/3, 3/4, 4/1; ...

%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] = 2; a[1, 1] = 1; Flatten[ Table[ a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* _Jean-François Alcover_, Sep 28 2011 *)

%K nonn,easy,tabf

%O 1,1

%A _Wolfdieter Lang_, Oct 19 2001