Two new sequences (somewhat) related to the golden number.

[Michele Dondi]

...and a general question.

The general question is as follows: suppose I have an interesting 
{problem,theorem,argument,whatever} that leads to an interesting sequence. 
Now the sequence is conceptually simple, but it can't be given properly a 
really short description. What to do then? I may write a "_sort_ _of_ 
brief article" and upload it somewhere and refer to it, but it seems quite 
pretentious...

However this question was raised by the two sequences pasted at the end of 
this mail, that I submitted to OEIS this morning. (I tried to distribute 
the relevant content amongst the %N and %C lines.)

Here I'll sum up as briefly as possible the main points of the reasoning 
leading to them.

Consider the matricial equation

(1)  G^2 = G+1,

where 1=I is the unit matrix. Given the solutions g,g'=1-g of (1) 
interpreted as a _real_ equation (the golden number and minus its inverse, 
or vice versa, according to the convention), the matricial solutions of 
(1) are gI, g'I and G=G(B) given by

(2)  2G=[1+p q-B]
         [q+B 1-p],

with

(2')  5+B^2 = p^2 + q^2.

Now it is natural investigate _integer_ solutions of (2'), which lead to 
solutions with coefficients in Z/2.

It turns out that:

(i) B must be even (immediate),
(ii) for even B the relevant quantity of the problem is the number of 
solutions of (2') for

(3)  p,q>0,  2p^2<5+B^2,

(iii) given the number a(n) of such solutions for B=2n, then the number of 
solutions of (1) of the form (2) with entries in Z is

(4)  4a(1)=4 if n=1,
      8a(n)   otherwise.

One may wonder which is the "intrinsic" meaning of the sequence index, 
that is of B, for a given matrix satisfying (1). The answer is that its 
absolute value is in bijective correspondence with the (operatorial) norm 
of G by means of

(5)  |2G| = 2|G| = sqrt(1+B^2) + sqrt(5+B^2)

(incidentally |G(B=0)| = g, |G(B=2)| = g^2 = g+1)

which can be even inverted eplicitly giving

(5')  B=sqrt((x^2-3x+1)/x),  x=|G|^2,

although it is by no means apparent from the latter form that it is an 
integer, for G with integer entries.


The sequences:

%I A104767
%S A104767 1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,1,0,0,0,0,1,0,2,1,0,0,1,0,0,0,0,0,2,1,0,1,0,0,0,0,1,0,2,0,1,0,0,0,0,1,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,2,1,1,0,1,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,1,0,0,0,0,1,0,1,0,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0
%N A104767 Number of solutions to 5+B^2=p^2+q^2 with B=2n, p,q>0 and 2p^2<5+B^2.
%C A104767 The number of matricial solutions with entries in Z to G^2=G+1 not of the form gI or g'I (g, the golden number, and g'=1-g are the solutions to x^2=x+1), hence of the form (1+p q-B \ q+B 1-p) with p^2+q^2=5+B^2 is given by 8a(n) for n!=1 and by 4a(1)=4 for n=1.
%e A104767 a(0)=1 because 5+0^2=5=1^2+2^2. a(15)=2 because 5+30^2=905=8^2+29^2=11^2+28^2.
%Y A104767 Cf. 104768.
%O A104767 0
%K A104767 ,easy,nice,nonn,
%A A104767 Michele Dondi (blazar at lcm.mi.infn.it), Mar 24 2005

%I A104768
%S A104768 8,4,0,8,0,0,8,0,8,0,8,0,0,0,0,16,0,0,8,0,0,16,0,0,8,0,0,0,8,0,0,0,0,8,0,16,8,0,0,8,0,0,0,0,0,16,8,0,8,0,0,0,0,8,0,16,0,8,0,0,0,0,8,8,0,0,16,0,0,0,0,0,0,0,0,0,0,0,16,0,16,8,8,0,8,0,0,0,0,0,16,8,0,0,0,0,0,0,0,8,0,0,8,0,0,16,0,0,16,8,0,0,0,0,8,0,8,0,0,0,32,0,0,8,0,0,16,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,16,16,0,0,8,0,0,0,0,0
%N A104768 Number of matricial solutions with entries in Z to G^2=G+1 of the form 2G=[1+p q-2n \ q+2n 1-p].
%C A104768 The matricial solutions to G^2=G+1 are gI, g'I (g is the golden number and g'=1-g) and the matrices 2G=[1+p q-B \ q+B 1-p]. It's easy to see that B must be even.
%F A104768 a(n)=8*104767(n) if n != 1, a(1)=4.
%Y A104768 Cf. 104767
%O A104768 0
%K A104768 ,easy,nice,nonn,
%A A104768 Michele Dondi (blazar at lcm.mi.infn.it), Mar 24 2005


Michele
-- 
I once heard someone say, the sum of the reciprocals 
of the known primes is less than 4 - and it will always be.
- Gerry Myerson in sci.math, "Re: Summation of Reciprocals of Primes"