When Does A083952(n)=1?

[Paul D Hanna]

Consider the series defined by A083952:
Sequence:  
  1,2,1,2,2,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,
  1,2,1,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,1,2,2,2,
  1,2,1,2,2,2,2
Name:   
  Integer coefficients of A(x), where 1<=a(n)<=2, such that
  A(x)^(1/2) consists entirely of integer coefficients.
Comments:  
  More generally, "integer coefficients of A(x), where 1<=a(n)<=m, 
  such that A(x)^(1/m) consists entirely of integer coefficients", 
  appears to have a unique solution for all m>0. 

The coefficients that have a value of 1 are the powers of x given by: 
  a(k)=1 at k=0,2,6,10,12,26,30,32,36,50,52,56,60,62,...

and the next power exceeds 99 ... 

What are the next several powers of x that have a coefficient of 1?

Would appreciate it if anyone could supply more of these unit terms.

Thanks Much, 
  Paul
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Final comments.

A(x)^(1/2) consists of the following integer coefficients:

1,1,0,1,0,1,-1,2,-2,4,-6,10,-16,27,-44,75,-127,218,-375,650,
-1130,1974,-3460,6086,-10736,18993,-33685,59882,-106683,190446,
-340611,610243,-1095102,1968200,-3542468,6384518,-11521308,20815942,
-37651528,68176596,-123574852,224204708,-407153894,740035534,
-1346197858,2450817682,-4465226160,8141263920,-14853972784,
27119593696,-49545204486,90570508650,-165664031716,303190783870,
-555188918444,1017170603474,-1864518543488,3419413588336,
-6273950961304,11516701084178,-21149779661384,...

The limiting ratio of the coefficients above should be 
the zero of A(x):
   A(x)=0 at x=-0.530852489019085 (approximately).