A038762 a(n) = 6*a(n-1) - a(n-2) for n >= 2, with a(0)=3, a(1)=13.

3, 13, 75, 437, 2547, 14845, 86523, 504293, 2939235, 17131117, 99847467, 581953685, 3391874643, 19769294173, 115223890395, 671574048197, 3914220398787, 22813748344525, 132968269668363, 774995869665653, 4517006948325555, 26327045820287677, 153445267973400507

Offset

0,1

Comments

This gives part of the (increasingly sorted) positive solutions x to the Pell equation x^2 - 2*y^2 = +7. For the y solutions see A038761. The other part of solutions is found in A101386 and A253811. - Wolfdieter Lang, Feb 05 2015

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

M. J. DeLeon, Pell's Equation and Pell Number Triples, Fib. Quart., 14(1976), pp. 456-460.

Tanya Khovanova, Recursive Sequences

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

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = sqrt(2*(A038761(n))^2+7).

a(n) = (13*((3+2*sqrt(2))^n -(3-2*sqrt(2))^n)-3*((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)).

a(n) = A077443(2n) = A038725(n)+A038725(n+1).

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3); a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^(n-1)+(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^(n-1). - Antonio Alberto Olivares, Apr 20 2008

G.f.: (3-5*x)/(1-6*x+x^2). - Philippe Deléham, Nov 03 2008, corrected by R. J. Mathar, Nov 06 2011

a(n) = -5*A001109(n) +3*A001109(n+1). - R. J. Mathar, Nov 06 2011

a(n) = rational part of z(n) = (3 + sqrt(2))*(3 + 2*sqrt(2))^n, n >= 0. z(n) gives only one part of the positive solutions to the Pell equation x^2 - 2*y^2 = 7. See the Nagell reference on how to find z(n), and a comment above. - Wolfdieter Lang, Feb 05 2015

E.g.f.: exp(3*x)*(3*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Mar 16 2024

Example

a(3)^2 - 2*A038761(3)^2 = 437^2 - 2*309^2 = +7. - Wolfdieter Lang, Feb 05 2015

Mathematica

LinearRecurrence[{6, -1}, {3, 13}, 40] (* Vincenzo Librandi, Nov 16 2011 *)

Prog

(Magma) I:=[3, 13]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
(PARI) x='x+O('x^30); Vec((3-5*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jul 26 2018

Crossrefs

Cf. A001109, A038725, A038761, A077443, A101386, A253811.

Sequence in context: A333890 A009382 A110193 * A276894 A074517 A251658

Adjacent sequences: A038759 A038760 A038761 * A038763 A038764 A038765

Keyword

easy,nonn

Author

Barry E. Williams, May 03 2000

Extensions

More terms from James A. Sellers, May 04 2000

Unspecific Pell comment replaced by Wolfdieter Lang, Feb 05 2015

Status

approved