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A077445 Numbers n such that (n^2 - 8)/2 is a square. 7
4, 20, 116, 676, 3940, 22964, 133844, 780100, 4546756, 26500436, 154455860, 900234724, 5246952484, 30581480180, 178241928596, 1038870091396, 6054978619780, 35291001627284, 205691031143924, 1198855185236260 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The equation "(n^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(a(n+1)*z^n,n=0..infinity) is the formal power series for tan(4*x)/tan(x) if one puts

  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 - 6xy + 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

Eric Weisstein's World of Mathematics, Pell Equation

Index entries for linear recurrences with constant coefficients, signature (6,-1).

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+2sqrt(2), b=3-2sqrt(2): a(n) = sqrt(2)(a^((2n-1)/2)+b^((2n-1)/2)). a(n) = sqrt(2*A003499(2n-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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A001653, A003499, A075870.

Sequence in context: A258664 A231539 A106567 * A085458 A085456 A120915

Adjacent sequences:  A077442 A077443 A077444 * A077446 A077447 A077448

KEYWORD

nonn,easy

AUTHOR

Gregory V. Richardson, Nov 09 2002

STATUS

approved

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Last modified November 20 10:31 EST 2017. Contains 294963 sequences.