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

Author

Barry E. Williams, May 03 2000

Keywords

• easy
• nonn

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

Examples

			a(3)^2 - 2*A038761(3)^2 = 437^2 - 2*309^2 = +7. - _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
• Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 55, 56.
• 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.

Crossrefs

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

Programs

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

Mathematica

LinearRecurrence[{6,-1},{3,13},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

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

Extensions

More terms from James Sellers, May 04 2000

Unspecific Pell comment replaced by Wolfdieter Lang, Feb 05 2015