A064883 - OEIS

%I #20 Jul 31 2023 18:29:48

%S 1,3,1,4,3,1,5,4,7,3,1,6,5,9,4,11,7,10,3,1,7,6,11,5,14,9,13,4,15,11,

%T 18,7,17,10,13,3,1,8,7,13,6,17,11,16,5,19,14,23,9,22,13,17,4,19,15,26,

%U 11,29,18,25,7,24,17,27,10

%N Eisenstein array Ei(1,3).

%C In Eisenstein's notation this is the array for m=1 and n=3; 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 n is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 4*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 1/3. The composition rule for 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)=1, a(1, 1)=3.

%e {1,3}; {1,4,3}; {1,5,4,7,3}; {1,6,5,9,4,11,7,10,3}; ...

%e This binary subtree of rationals is built from 1/3; 1/4, 4/3; 1/5, 5/4, 4/7, 7/3; ...

%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] = 3; Flatten[ Table[ a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* _Jean-François Alcover_, Oct 03 2011 *)

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Oct 19 2001