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A054488 Expansion of (1+2x)/(1-6x+x^2). 9
1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Bisection (even part) of Chebyshev sequence with Diophantine property.

b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).

The odd part is A077413(n) with Diophantine companion A077239(n).

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 sequences related to Chebyshev polynomials.

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

FORMULA

a(n) = 6*a(n-1)-a(n-2), a(0)=1, a(1)=8.

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

a(n) = S(n, 6)+2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the second kind, A049310. S(n, 6)= A001109(n+1).

a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-9)^k. - Philippe Deléham, Mar 05 2014

EXAMPLE

8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.

MAPLE

a[0]:=1: a[1]:=8: 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

CROSSREFS

Cf. A002315 and A038761.

A077241 (even and odd parts).

Sequence in context: A051140 A255720 A014524 * A034349 A024108 A247726

Adjacent sequences:  A054485 A054486 A054487 * A054489 A054490 A054491

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, May 04 2000

EXTENSIONS

More terms from James A. Sellers, May 05 2000

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.