Series of Square Coefficients

Paul D Hanna pauldhanna at juno.com
Sat Apr 26 17:19:03 CEST 2003

```Consider a power series A(x) with positive integer coefficients
such that the coefficients of
A(x)^2 + A(x) - 1
consist entirely of squares.

An example of this is found in A083352, which is to be the least case.
Are there other, more interesting solutions?

What is somewhat surprising to me is that the terms (after the initial 1)
of A085332 all seem to be multiples of 3.

In fact, the resultant series of square coefficients (after the initial
1)
of A083353 also seem to be multiples of 9.

These sequences are paste below.

Could someone extend these sequences?  It would be interesting if there
were
any pattern to the terms (especially for A083353).

Thanks,
Paul D Hanna
----------------------------------------------------------------------
%S A083352 1,3,9,9,3,15,33,18,36,24,75,96,51,96,159,165,255,168,27,60,
333,255,66,18,441,291,735,258,390,696,480,696,1062,
%N A083352 Least positive integer coefficients of power series A(x) such
that
the coefficients of A(x)^2 + A(x) - 1 consist entirely of
squares.
%C A083352 After the first term, all terms seem to be a multiple of 3.
%e A083352 A(x)=1 + 3x + 9x^2 + 9x^3 + 3x^4 + 15x^5 + 33x^6 +...;
so that A083353(x) =
A(x)^2 + A(x) - 1 = 1 + 9x + 36x^2 + 81x^3 + 144x^4 + 225x^5 + 324x^6
+...

For n>0, A083352(n)/3 =

1,3,3,1,5,11,6,12,8,25,32,17,32,53,55,85,56,9,20,
111,85,22,6,147,97,245,86,130,232,160,232,354,...

----------------------------------------------------------------------

%S A083353 1,9,36,81,144,225,324,576,1089,1296,1764,2916,4356,6084,7569,
9801,16641,20736,25281,32400,39204,53361,69696,76176,90000,110889,149769,
176400,184041,207936,281961,338724,363609,
%N A083353 Coefficients of power series A(x) consist entirely of squares,

where A(x) = A083352(x)^2 + A083352(x) - 1.
%C A083353 After the first term, all terms seem to be a multiple of 9.

For n>0, sqrt( A083353(n) )/3 =

1,2,3,4,5,6,8,11,12,14,18,22,26,29,33,43,48,53,60,
66,77,88,92,100,111,129,140,143,152,177,194,201,...
----------------------------------------------------------------------

```