|
| |
|
|
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; 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
| I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7(1969), pps. 181-193.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pps. 122-125, 194-196.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7(1969), pps. 231-242.
|
|
|
LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
|
|
|
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 (creighton.k.dement(AT)uni-oldenburg.de), Nov 28 2004. Comment from Vim Wenders (vim(AT)gmx.li), 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)).
|
|
|
CROSSREFS
| Cf. A001353, A001834.
Sequence in context: A078798 A027043 A006815 * A013261 A013265 A038383
Adjacent sequences: A054488 A054489 A054490 * A054492 A054493 A054494
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Barry E. Williams, May 04 2000
|
|
|
EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 05 2000
Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
|
| |
|
|
