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A054491 a(n) = 4a(n-1) - a(n-2), a(0)=1, a(1)=6. 10
1, 6, 23, 86, 321, 1198, 4471, 16686, 62273, 232406, 867351, 3236998, 12080641, 45085566, 168261623, 627960926, 2343582081, 8746367398, 32641887511, 121821182646, 454642843073, 1696750189646, 6332357915511, 23632681472398 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

The odd part is A077234(n) with Diophantine companion A077235(n).

-3*a(n)^2 + b(n)^2 = 13, with the companion sequence b(n)= A077236(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..23.

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 (4,-1).

FORMULA

a(n) = (6*((2+sqrt(3))^n-(2-sqrt(3))^n)-((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)))/(2*sqrt(3)).

a(n) = 6*S(n-1, 4) - S(n-2, 4) = S(n, 4) + 2*S(n-1, 4), with S(n, x) := U(n, x/2) Chebyshev's polynomials of 2nd kind, A049310. S(-1, x) := 0, S(-2, x) := -1, S(n, 4)= A001353(n+1).

G.f.: (1+2*x)/(1-4*x+x^2).

Conjecture: a(n+1) = A001353(n+2) + 2*A001353(n+1) - Creighton Dement, Nov 28 2004. Comment from Vim Wenders, Mar 26 2008: The conjecture is easily verified using a(n) = (6*((2+sqrt(3))^n-(2-sqrt(3))^n)-((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)))/(2*sqrt(3)) and A001353(n)=[(2+sqrt(3))^n-(2-sqrt(3))^n]/(2*sqrt(3)).

a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-7)^k. - Philippe Deléham, Mar 05 2014

PROG

(PARI) a(n) = if (n==0, 1, if (n==1, 6, 4*a(n-1)-a(n-2))) \\ Michel Marcus, Jun 23 2013

CROSSREFS

Cf. A001353, A001834.

Sequence in context: A006815 A264690 A241394 * A282710 A295132 A013261

Adjacent sequences:  A054488 A054489 A054490 * A054492 A054493 A054494

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, May 04 2000

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified November 20 00:42 EST 2017. Contains 294957 sequences.