1,1

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

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, p. 341-400.

Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

J. J. O'Connor and E. F. Robertson, Pell's Equation

Eric Weisstein's World of Mathematics, ; Pell Equation

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)

G.f.: 4(x-x^2)/(1-6x+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) = 7*a(n-1) - 7*a(n-2) + a(n-3); a(0) = 4, a(1) = 20, a(2) = 116; a(n) = (2 + SQRT(2))*(3 + 2*SQRT(2))^n + (2 - SQRT(2))*(3- 2*SQRT(2))^n - Antonio A. Olivares (olivares14031(AT)gmail.com), Feb 23 2006

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

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

Adjacent sequences: A077442 A077443 A077444 this_sequence A077446 A077447 A077448

Sequence in context: A128327 A100034 A106567 this_sequence A085458 A085456 A120915

nonn

Gregory V. Richardson (omomom(AT)hotmail.com), Nov 09 2002