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A038723 a(n) = 6a(n-1) - a(n-2), n >= 2, a(0)=1, a(1)=4. 13
1, 4, 23, 134, 781, 4552, 26531, 154634, 901273, 5253004, 30616751, 178447502, 1040068261, 6061962064, 35331704123, 205928262674, 1200237871921, 6995498968852, 40772755941191, 237641036678294, 1385073464128573 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence gives one half of all positive solutions y = y1 = a(n) of the first class of the Pell equation x^2 - 2*y^2 = -7. For the corresponding x=x1 terms see A054490(n). Therefore it gives also one fourth of all positive solutions x = x1 of the first class of the Pell equation x^2 - 2*y^2 = 14, with the y=y1 terms given by A054490. - Wolfdieter Lang, Feb 26 2015

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 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) = ((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

a(n) = A001653(n+1) - A001109(n). - 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 - 2*x) / (1 - 6*x + x^2). a(n) = (7 + a(n-1)^2) / a(n-2). - Michael Somos, Sep 28 2008

a(n) = Sum_{k = 0..n} A238731(n,k)*3^k. - Philippe Deléham, Mar 05 2014

a(n) = S(n,6) - 2*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 comment above. - Wolfdieter Lang, Feb 26 2015

a(0) = -(A038761(0) - A038762(0))/2, a(n) = (A253811(n-1) + A101386(n-1))/2, n >= 1. See the Mar 19 2015 comment on A054490. - Wolfdieter Lang, Mar 19 2015

EXAMPLE

n = 2: A054490(2)^2 - 2*(2*a(2))^2 =

       65^2 - 2*(2*23)^2 = -7,

      (4*a(2))^2 - 2*A054490(2)^2 =

      (4*23)^2 - 2*65^2 = 14. - Wolfdieter Lang, Feb 26 2015

a(2) = (A253811(1) + A101386(1))/2 = (19 + 27)/2 = 23. - Wolfdieter Lang, Mar 19 2015

MAPLE

a[0]:=1: a[1]:=4: 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

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, A001541, A038725, A054490, A001109.

A038725(n) = a(-n).

Sequence in context: A227639 A239399 A193808 * A091640 A237362 A067110

Adjacent sequences:  A038720 A038721 A038722 * A038724 A038725 A038726

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, May 02 2000

EXTENSIONS

More terms from James A. Sellers, May 03 2000

STATUS

approved

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Last modified June 27 15:07 EDT 2017. Contains 288790 sequences.