A038725 a(n) = 6*a(n-1) - a(n-2), n >= 2, a(0)=1, a(1)=2.

1, 2, 11, 64, 373, 2174, 12671, 73852, 430441, 2508794, 14622323, 85225144, 496728541, 2895146102, 16874148071, 98349742324, 573224305873, 3340996092914, 19472752251611, 113495517416752, 661500352248901, 3855506596076654, 22471539224211023, 130973728749189484

Offset

0,2

Comments

From Wolfdieter Lang, Feb 26 2015: (Start)

The sequence {2*a(n+1)}_{n >= 0}, gives all positive solutions y = y2(n) = 2*a(n+1) of the second class of the Pell equation x^2 - 2*y^2 = -7. For the corresponding terms x = x2(n) see A255236(n).

See A255236 for comments on the first class solutions and the relation to the Pell equation x^2 - 2*y^2 = 14. (End)

8*a(n)^2 - 7 is a perfect square. (x, y) = (a(n), a(n+1)) are solutions of ((x + y)^2 - 1)/(x*y - 1) = 8. The other solutions are in A038723. - Klaus Purath, Nov 17 2025

References

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

Links

Stefano Spezia, Table of n, a(n) for n = 0..1300

I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.

Seyed Hassan Alavi, Ashraf Daneshkhah, and Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See x'(n) in Lemma 7.9 p. 21.

Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 55, 56.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.

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) = 7*a(n-1) - 7*a(n-2) + a(n-3); a(n) = ((4-sqrt(2))/8)*(3+2*sqrt(2))^(n-1)+((4+sqrt(2))/8)*(3-2*sqrt(2))^(n-1). - Antonio Alberto Olivares, Mar 29 2008

From Michael Somos, Sep 28 2008: (Start)

Sequence satisfies -7 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 6*u*v.

G.f.: (1 - 4*x) / (1 - 6*x + x^2). a(n) = (7 + a(n-1)^2) / a(n-2). (End)

From Wolfdieter Lang, Feb 26 2015: (Start)

a(n) = S(n, 6) - 4*S(n-1, 6), n>=0, with the Chebyshev polynomials S(n, x) (A049310), with S(-1, x) = 0, evaluated at x = 6. S(n, 6) = A001109(n-1). See the g.f. and the Pell equation comment above.

a(n) = 6*a(n-1) - a(n-2), n >= 1, a(-1) = 4, a(0) = 1. (See the name.) (End)

From Wolfdieter Lang, Mar 19 2015: (Start)

a(n+1) = sqrt((A255236(n)^2 + 7)/2)/2, n >= 0.

a(n+1) = (A038761(n) + A038762(n))/2, n >= 0. See the Mar 19 2015 comment on A054490. (End)

E.g.f.: exp(3*x)*(4*cosh(2*sqrt(2)*x) - sqrt(2)*sinh(2*sqrt(2)*x))/4. - Stefano Spezia, May 01 2020

0 = a(n+1)*a(n-1) - a(n)^2 - 7 = a(n)^2 + a(n+1)^2 - 6*a(n)*a(n+1) + 7 for all n in Z. - Michael Somos, Mar 04 2026

Example

n = 2: a(3) = sqrt((181^2 + 7)/2)/2 = 64.
a(3) = (53 + 75)/2 = 64. - Wolfdieter Lang, Mar 19 2015

Maple

a[0]:=1: a[1]:=2: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006

Mathematica

Union[Flatten[NestList[{#[[2]], #[[3]], 6#[[3]]-#[[2]]}&, {1, 2, 11}, 25]]]  (* Harvey P. Dale, Mar 04 2011 *)
LinearRecurrence[{6, -1}, {1, 2}, 30] (* Harvey P. Dale, Jun 12 2017 *)
a[ n_] := ChebyshevT[n, 3] - ChebyshevU[n-1, 3]; (* Michael Somos, Mar 04 2026 *)

Prog

(PARI) {a(n) = real((3 + 2*quadgen(8))^n * (1 - quadgen(8) / 4))} /* Michael Somos, Sep 28 2008 */
(PARI) {a(n) = polchebyshev(n, 1, 3) - polchebyshev(n-1, 2, 3)} /* Michael Somos, Sep 28 2008 */

Crossrefs

Cf. A001653 and A001541. Cf. A001109.

A038723(n) = a(-n).

Sequence in context: A362799 A179120 A373890 * A386006 A161947 A382918

Adjacent sequences: A038722 A038723 A038724 * A038726 A038727 A038728

Keyword

easy,nonn

Author

Barry E. Williams, May 02 2000

Status

approved