- Dear friends,

Is there anybody on the list who can tell me how the sequences

http://www.research.att.com/~njas/sequences/A004253

http://www.research.att.com/~njas/sequences/A004254

are derived in the paper

F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C.

Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

Kind regards,

Floor. - At 5.44 PM +0200 29-4-06, Floor en Lyanne van Lamoen wrote:
>Dear friends,

That book was a Festschrift for the 80th birthday of H. G.

>

>Is there anybody on the list who can tell me how the sequences

>http://www.research.att.com/~njas/sequences/A004253

>http://www.research.att.com/~njas/sequences/A004254

>are derived in the paper

>F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C.

>Butcher, editor, A Spectrum of Mathematics. Auckland University

>Press, 1971....

Forder. I'm sorry I don't know who Frank A. Haight was.

Haight starts with a right-angled triangle with sides a, b, c

(the hypotenuse), and defines a succession of squares outside it.

He refers briefly to each square by the length of its side, e.g.

"square b" means the square in the diagram having side b. He writes

a_1 = a, b_1 = b, c_1 = c, then (in his own words):

"For j = 1, 2, 3, ...

x_j is constructed on the segment joining adjacent corners of b_j and c_j

y_j is constructed on the segment joining adjacent corners of a_j and c_j

z_j is constructed on the segment joining adjacent corners of b_j and a_j

For j = 2, 3, ...

a_j is constructed on the segment joining adjacent corners of

y_(j-1) and z_(j-1)

b_j is constructed on the segment joining adjacent corners of

x_(j-1) and z_(j-1)

c_j is constructed on the segment joining adjacent corners of

x_(j-1) and y_(j-1)

In the above definitions the 'adjacent corners' referred to are those

'exterior to the diagram' (see Fig. 1), which are not already

connected, and such that the connection would not pass through the

figure."

Haight then names some angles and derives several formulae

leading to the sequences which you asked about. The ratios 1 : 4

: 19 : 91 : ....

= a_1 : a_2 : a_3 : a_4 : ....

= b_1 : b_2 : b_3 : b_4 : ....

= c_1 : c_2 : c_3 : c_4 : ....

and the ratios 1 : 5 : 24 : 115 : ....

= x_1 : x_2 : x_3 : x_4 : ....

= y_1 : y_2 : y_3 : y_4 : ....

= z_1 : z_2 : z_3 : z_4 : ....

Of course it's easier to follow with the diagram. If you'd

like to send your snail-mail address I can easily post you a

photo-copy of this little article.

Ken Pledger.