%I #30 Jul 31 2023 18:29:28

%S 3,1,3,4,1,3,7,4,5,1,3,10,7,11,4,9,5,6,1,3,13,10,17,7,18,11,15,4,13,9,

%T 14,5,11,6,7,1,3,16,13,23,10,27,17,24,7,25,18,29,11,26,15,19,4,17,13,

%U 22,9,23,14,19,5,16,11,17,6,13

%N Eisenstein array Ei(3,1).

%C In Eisenstein's notation this is the array for m=3 and n=1; see pp. 41-42 of the Eisenstein reference given for A064881. This is identical with the array for m=1, n=3, given in A064883, 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 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 3/1. 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)=3, a(1, 1)=1.

%e Array begins

%e {3, 1};

%e {3, 4, 1};

%e {3, 7, 4, 5, 1};

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

%e This binary subtree of rationals is built from

%e 3/1;

%e 3/4, 4/1;

%e 3/7, 7/4, 4/5, 5/1; ...

%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] = 3; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* _Jean-François Alcover_, Sep 28 2011 *)

%t eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;

%t Flatten@NestList[eisen, {3, 1}, 6] (* _Harlan J. Brothers_, Feb 18 2015 *)

%K nonn,easy,tabf

%O 1,1

%A _Wolfdieter Lang_, Oct 19 2001