0,2

All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not square-free.

Tanya Khovanova, Recursive Sequences

Index entries for sequences relate d to Chebyshev polynomials.

a(n)= S(n, 27)=U(n, 27/2)= S(2*n+1, sqrt(29))/sqrt(29) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).

a(n)=27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0.

a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (27+5*sqrt(29))/2 and am := (27-5*sqrt(29))/2.

G.f.: 1/(1-27*x+x^2).

(x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4.

with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008

sage: [lucas_number1(n, 27, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

Sequence in context: A095898 A014914 A157461 this_sequence A073537 A016947 A167726

Adjacent sequences: A097778 A097779 A097780 this_sequence A097782 A097783 A097784

nonn,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004