[seqfan] re A064098 (Markov equation)

[wouter meeussen]

hi all,

would it be worth noting that A064098, apart from its 3 initial 1's,
is not only a subsequence of the Markov numbers (A002559),
but precisely the largest integer produced at each generation?
Here I define 'generation' as implicit in 'The book of Numbers' (J.H.
Conway, R.K. Guy, pg 188).

It might also be nice to enter the (2^n+1) Markov numbers in this unsorted,
natural order (*1*):
{1, 2},
{1, 5, 2},
{1, 13, 5, 29, 2},
{1, 34, 13, 194, 5, 433, 29, 169, 2},
{1, 89, 34, 1325, 13, 7561, 194, 2897, 5, 6466, 433, 37666, 29, 14701, 169,
985, 2}

or, more economically, only the newcomers (nth row with 2^n  elements) (*2*)
{5},
{13, 29},
{34, 194, 433, 169},
{89, 1325, 7561, 2897, 6466, 37666, 14701, 985},
{233, 9077, 135137, 51641, 294685, 4400489, 1686049, 43261,
       96557, 8399329, 48928105, 3276509, 1278818, 7453378, 499393, 5741}

the positions of those A064098 maximal elements are
in (*1*) : 2,2,4,6,12,22,44,86,172,342 ;  A128209 Jacobsthal numbers
(A001045)+1.
in (*2*) : 1,2,3,6,11,22,43,86,171 ; A005578: a(2n)=2a(2n-1),
a(2n+1)=2a(2n)-1.

but maybe all this is old hat,

Wouter.