1,2

a(n) gives minimal (positive) solution of Pell equation a(n)^2 - D(n)*b(n)^2 = -4 with D(n)=A077426(n). The companion sequence is b(n)=A078357(n).

Computed from Perron's table (see reference p. 108) which gives the minimal x,y values for the Diophantine eq. x^2 - x*y - ((D(m)-1)/4)*y^2= +1, resp., -1 if D(m)=A077425(m), resp, D(m)=A077425(m) and D(m) in A077426 (this second case excludes in Perron's table the D values with a 'Teilnenner' in brackets).

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 see a comment in A077428. Here only D values with no 'Teilnenner' in brackets are of interest and a(n)=2*x(n)-y(n) and b(n)=y(n). E.g. D=41, with 'Teilnenner von (sqrt(D)+1)/2' in the notation, explained in an example of A077427, 3,1,2 (period length k=5) and (x,y)=(37,10) which translates to the minmal solution (a,b)=(64,10).

Generic D(n) values are those from A078370(k)=(4*k(k+1)+5), k>=0, which are 5 (mod 8). For such D values the minimal solution is (a,b)=(2*k+1,1) (e.g. D(7)= A077426(7) = 53 = A078370(3) with a(7)= 2*3+1=7 and b(7)=A078357(7)=1).

The general solution of Pell a^2-D(n)*b^2 = -4 with generic D(n)=A078370(k), k>=0, is a(n,m)= (2*k+1)*S(2*m,sqrt(D(n))) and b(n,m)= T(2*m+1,sqrt(D(n))/2)/(sqrt(D(n))/2), 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 A078370) the general solution of a^2-D(n)*b^2 = -4 is a(n,m)=a(n)*S(2*m,sqrt(a(n)^2+4)) and b(n,m)= b(n)*T(2*m+1,sqrt(a(n)^2+4)/2)/(sqrt(a(n)^2+4)/2), 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).

Index entries for sequences related to Chebyshev polynomials.

41=D(6)=A077426(6) (also A077425(8)), hence a(6)=64 and b(6)=A078357(6)=10 satisfies 64^2 - 41*10^2 = -4.

Sequence in context: A155676 A077151 A058055 this_sequence A050093 A120072 A166492

Adjacent sequences: A078353 A078354 A078355 this_sequence A078357 A078358 A078359

nonn

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

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 24 2009