1,1

Computed from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values for the Diophantine eq. x^2 - x*y - ((D(n)-1)/4)*y^2= +1, resp., -1 if D(n)=A077425(n), resp, D(n)=A077425(n) and D(n) also in A077426.

The conversion from the x,y values of Perron's table to the minimal a=a(n) and b=b(n) solutions of a^2 - D(n)*b^2 =+4 is as follows. If D(n)=A077425(n) but not from A077426 (period length of continued fraction of (sqrt(D(n))+1)/2 is even) then a(n)=2*x(n)-y(n) and b(n)=y(n). E.g. D(4)=21 with Perron's (x,y)=(3,1) and (a,b)=(5,1). 1=b(4)=A078355(4). If D(n)=A077425(n) appears also in A077426 (odd period length of continued fraction of (sqrt(D(n))+1)/2) then a(n)=(2*x-y)^2+2 and b(n)=(2*x-y)*y. E.g. D(7)=37 with Perron's (x,y)=(7,2) leading to (a,b)=(146,24) with 24=b(7)=A078355(7).

The generic D(n) values are those from A078371(k-1) := (2*k+3)*(2*k-1), for k>=1, which are 5 (mod 8). For such D values the minimal solution is (a(n),b(n))=(2*k+1,1) (e.g. D(16)=77= A078371(3) with a(16)=2*4+1=9 and b(16)=A078355(16)=1).

The general solution of Pell a^2-D(n)*b^2 = +4 with generic D(n)=A077425(n)=A078371(k-1), k>=1, is a(n,m)= 2*T(m+1,(2*k+1)/2) and b(n,m)= S(m,2*k+1), m>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 resp. A049310.

For non-generic D(n) (not from A078371) the general solution of a^2-D(n)*b^2 = +4 is a(n,m)= 2*T(m+1,a(n)/2) and b(n,m)= b(n)*S(m,a(n)), m>=0, with Chebyshev's polynomials and in this case b(n)>1.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

S. R. Finch, Class number theory

Index entries for sequences related to Chebyshev polynomials.

Sequence in context: A096655 A030226 A132101 this_sequence A074504 A126115 A018193

Adjacent sequences: A077425 A077426 A077427 this_sequence A077429 A077430 A077431

nonn,more

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002