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%I #25 May 15 2022 12:32:36
%S 3,2,3,5,2,3,8,5,7,2,3,11,8,13,5,12,7,9,2,3,14,11,19,8,21,13,18,5,17,
%T 12,19,7,16,9,11,2,3,17,14,25,11,30,19,27,8,29,21,34,13,31,18,23,5,22,
%U 17,29,12,31,19,26,7,23,16
%N Eisenstein array Ei(3,2).
%C In Eisenstein's notation this is the array for m=3 and n=2; see pp. 41-2 of the Eisenstein reference given for A064881. The array for m=n=1 is A049456.
%C For n >= 1, the number of entries of row 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 3/2. 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) = 3, a(1, 1) = 2.
%e Triangle begins:
%e {3, 2};
%e {3, 5, 2};
%e {3, 8, 5, 7, 2};
%e {3, 11, 8, 13, 5, 12, 7, 9, 2};
%e ...
%e This binary subtree of rationals is built from 3/2; 3/5, 5/2; 3/8, 8/5, 5/7, 7/2; ...
%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] = 3; a[1, 1] = 2; Flatten[Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n - 1)}]] (* _Jean-François Alcover_, Sep 28 2011 *)
%Y Cf. A002487, A007051, A049456, A064881.
%K nonn,easy,tabf
%O 1,1
%A _Wolfdieter Lang_, Oct 19 2001
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