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A076631 Solve 2^n - 2 = 7(x^2 - x) + (y^2 - y) for (x,y) with x>0, y>0; a(n) = value of y.

%I A076631 #30 Aug 22 2024 11:54:38
%S A076631 1,2,3,1,6,5,7,16,3,29,34,24,91,44,138,225,51,500,399,601,1398,197,
%T A076631 2599,2992,2206,8189,3778,12600,20155,5045,45354,35265,55443,125972,
%U A076631 15087,236857,267030,206684,740743,327376,1154110,1808861,499359,4117080
%N A076631 Solve 2^n - 2 = 7(x^2 - x) + (y^2 - y) for (x,y) with x>0, y>0; a(n) = value of y.
%C A076631 Euler (unpublished) showed there is a unique positive solution (x,y) for every positive n.
%C A076631 Proof of uniqueness: Let R be the ring of integers of Q(sqrt(-7)) and let a=(1+sqrt(-7))/2, b=(1-sqrt(-7))/2. It is easy to see that any element of R of even norm (=squared absolute value) can be divided by one of a or b to get back an element of R. Thus since a*b=2, the only elements in R of norm 2^n and of the form (p+q*sqrt(-7))/2 with p,q odd are a^n, b^n, -a^n, -b^n - precisely one of which lies in the first quadrant. Finally apply Dean Hickerson's remarks. - _Paul Boddington_, Jan 23 2004
%D A076631 A. Engel, Problem-Solving Strategies, Springer-Verlag, New York, 1998.
%F A076631 Note that the equation is equivalent to 2^(n+2) = (2y-1)^2 + 7 (2x-1)^2, so it is related to norms of elements of the ring of integers in the quadratic field Q(sqrt(-7)) and Euler's claim presumably follows from unique factorization in that field. From this we can get a formula for the x's and y's: Let a(n) and b(n) be the unique rational numbers such that a(n) + b(n) sqrt(-7) = ((1 + sqrt(-7))/2)^n. I.e., a(n) = (((1 + sqrt(-7))/2)^n + ((1 - sqrt(-7))/2)^n)/2. - _Dean Hickerson_, Oct 19 2002
%F A076631 a(n) = 2^(n/2)*abs(cos(n*t))+1/2, where t=arctan(sqrt(7)). - _Paul Boddington_, Jan 23 2004
%t A076631 Table[Reduce[{2^n-2==7(x^2-x)+(y^2-y),x>0,y>0},{x,y},Integers][[-1,-1]], {n,50}] (* _Harvey P. Dale_, Dec 15 2018 *)
%o A076631 (PARI) p(n,x,y)=2^n-2-7*(x^2-x)-(y^2-y)
%o A076631 a(n)=if(n<0,0,y=1; while(frac(real(component(polroots(p(n,x,y)),2)))>0,y++); y)
%Y A076631 Cf. A076632 (values of x).
%K A076631 nonn,easy
%O A076631 1,2
%A A076631 _Ed Pegg Jr_, Oct 17 2002
%E A076631 More terms from _Benoit Cloitre_, Oct 24 2002
%E A076631 More terms from Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 14 2005
%E A076631 Definition corrected by _Harvey P. Dale_, Dec 15 2018