A077445 Numbers k such that (k^2 - 8)/2 is a square.

4, 20, 116, 676, 3940, 22964, 133844, 780100, 4546756, 26500436, 154455860, 900234724, 5246952484, 30581480180, 178241928596, 1038870091396, 6054978619780, 35291001627284, 205691031143924, 1198855185236260

Offset: 1

Author

Gregory V. Richardson, Nov 09 2002

Keywords

• nonn
• easy

Comments

The equation "(k^2 - 8)/2 is a square" is a version of the generalized Pell Equation "x^2 - D*y^2 = C".

a(n)^2 - 2*A077444(n) = 8.

From Wolfdieter Lang, Jan 18 2013: (Start)

4*(1-z)/(1-6*z+z^2) = Sum_{n>=0} a(n+1)*z^n is the formal power series for tan(4*x)/tan(x) if one lets

z = (tan(x))^2. For the numerator and denominator of this o.g.f. see A034867 and A034839, respectively. Convergence holds for 0 <= z < 3 - 2*sqrt(2), approximately 0.1715728753. This means for |x| < Pi/8, approximately 0.3926990818.

See also the o.g.f. given by Johannes W. Meijer, Aug 01 2010, in the formula section of A001653 = (this sequence)/4.

(End)

Positive values of x (or y) satisfying x^2 - 6*x*y + y^2 + 64 = 0. - Colin Barker, Feb 13 2014

References

• A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
• L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
• Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J. J. O'Connor and E. F. Robertson, Pell's Equation
• Tanya Khovanova, Recursive Sequences
• Klaus Nagel, Length of cycles in bijection of 2 quadratic grids rotated by Pi/6 relative to each other, (2009).
• Eric Weisstein's World of Mathematics, Pell Equation
• Index entries for linear recurrences with constant coefficients, signature (6,-1).

Crossrefs

Cf. A001653, A003499, A075870.

Programs

Mathematica

CoefficientList[Series[4 (1 - x)/(1 - 6 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2014 *)

PARI

a(n)=if(n<1,0,subst(poltchebi(n)+poltchebi(n-1),x,3))

Formula

a(n) = (((3+2*sqrt(2))^n + (3-2*sqrt(2))^n) + ((3+2*sqrt(2))^(n-1) + (3-2*sqrt(2))^(n-1))) / 2.

a(n) = 6*a(n-1) - a(n-2) = 4*A001653(n).

G.f.: 4*(x-x^2)/(1-6*x+x^2).

With a=3+2*sqrt(2), b=3-2*sqrt(2): a(n) = sqrt(2)*(a^((2*n-1)/2) + b^((2*n-1)/2)). a(n) = sqrt(2*A003499(2*n-1)+4). - Mario Catalani (mario.catalani(AT)unito.it), Mar 24 2003

a(n) = (A003499(n+1) + A003499(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003

a(n) = (2 + sqrt(2))*(3 + 2*sqrt(2))^n + (2 - sqrt(2))*(3- 2*sqrt(2))^n. - Antonio Alberto Olivares, Feb 23 2006

a(n) = 2*A075870(n). - Bruno Berselli, Nov 27 2013

G.f.: 2*Q(0)*x*(1-x)/(1-3*x), where Q(k) = 1 + 1/( 1 - x*(8*k-9)/( x*(8*k-1) - 3/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2013