login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A038725 a(n) = 6a(n-1) - a(n-2), n >= 2, a(0)=1, a(1)=2. 10
1, 2, 11, 64, 373, 2174, 12671, 73852, 430441, 2508794, 14622323, 85225144, 496728541, 2895146102, 16874148071, 98349742324, 573224305873, 3340996092914, 19472752251611, 113495517416752, 661500352248901 (list; graph; refs; listen; history; text; internal format)
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)

REFERENCES

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

LINKS

Table of n, a(n) for n=0..20.

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

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

Sequence satisfies -7 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 6*u*v. - Michael Somos, Sep 28 2008

G.f.: (1 - 4*x) / (1 - 6*x + x^2). a(n) = (7 + a(n-1)^2) / a(n-2). - Michael Somos, Sep 28 2008

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. - Wolfdieter Lang, Mar 19 2015

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 *)

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: A080049 A126745 A179120 * A161947 A001565 A199412

Adjacent sequences:  A038722 A038723 A038724 * A038726 A038727 A038728

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, May 02 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 19:04 EDT 2018. Contains 315270 sequences. (Running on oeis4.)