# [seqfan] lower principal and upper principal

Creighton Kenneth Dement creighton.k.dement at mail.uni-oldenburg.de
Fri Apr 17 02:37:55 CEST 2009

```Dear Seqfans,

I'm not quite sure how "lower principal" and "upper principal" are defined
in the definitions of A143608 (Numerators of the lower principal and
intermediate convergents to 2^(1/2)) and A143609. Can someone please
confirm my g.f. conjecture for both sequences?

A143608 = Expansion of (1+x^2+4*x)/((x^2+2*x-1)*(x^2-2*x-1))

A143609 = Expansion of (2+3*x-2*x^2-x^3)/((x^2+2*x-1)*(x^2-2*x-1))

Wikipedia http://en.wikipedia.org/wiki/Continued_fraction defines
"semiconvergents, secondary convergents, or intermediate fractions" as
different words for the same concept and I assume that "intermediate
convergents" is related to this.

My program returns the following relationships among the above sequences.

mixseq: [1, 1, 7, 7, 41, 41, 239, 239, 1393, 1393, 8119, 8119, 47321]
2mixposseq: [1, 4, 7, 24, 41, 140, 239, 816, 1393, 4756, 8119, 27720, 47321]
2mixnegseq: [0, -3, 0, -17, 0, -99, 0, -577, 0, -3363, 0, -19601, 0]

mixseq = essentially NSW numbers
2mixposseq = A143608, Numerators of the lower principal and intermediate
convergents to 2^(1/2).
2mixnegseq = essentially A001541

Identity: mixseq = mixposseq + mixnegseq

2jesseq: [-1, -3, -3, -17, -17, -99, -99, -577, -577, -3363, -3363,
-19601, -19601]
2jesposseq: [1, 0, 7, 0, 41, 0, 239, 0, 1393, 0, 8119, 0, 47321]
2jesnegseq: [-2, -3, -10, -17, -58, -99, -338, -577, -1970, -3363, -11482,
-19601, -66922]

2jesseq = essentially A001541
2jesposseq = essentially NSW numbers
2jesnegseq = A143609, Numerators of the upper principal and intermediate
convergents to 2^(1/2).

Identity: jesseq = jesposseq + jesnegseq

Sincerely,
Creighton

```