|
| |
|
|
A075843
|
|
99*a(n)^2 + 1 is a square.
|
|
6
| |
|
|
0, 1, 20, 399, 7960, 158801, 3168060, 63202399, 1260879920, 25154396001, 501827040100, 10011386405999, 199725901079880, 3984506635191601, 79490406802752140, 1585823629419851199, 31636982181594271840
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*Sqrt(11).
Chebyshev's polynomials U(n,x) evaluated at x=10.
The a(n) give all (unsigned, integer) solutions of Pell equation b(n)^2 - 99*a(n)^2 = +1 with b(n)= A001085(n).
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 20's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imagianry unit). [From John M. Campbell, Jul 08 2011]
|
|
|
REFERENCES
| A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
|
|
|
LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for sequences related to Chebyshev polynomials.
|
|
|
FORMULA
| a(n) = [(10+3*Sqrt(11))^n - (10-3*Sqrt(11))^n] / (6*Sqrt(11))
a(n) = 20*a(n-1) - a(n-2), n>=1, a(0)=0, a(1)=1.
a(n) = S(n-1, 20), with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310.
G.f.: x/(1-20*x+x^2).
a(n) = sqrt((A001085(n)^2 - 1)/99).
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*19^k. - DELEHAM Philippe, Feb 10 2012
|
|
|
MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 10]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
|
|
|
PROG
| sage: [lucas_number1(n, 20, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
|
|
|
CROSSREFS
| Cf. A001084.
Sequence in context: A170739 A158534 A171325 * A090051 A089957 A207127
Adjacent sequences: A075840 A075841 A075842 * A075844 A075845 A075846
|
|
|
KEYWORD
| nonn,changed
|
|
|
AUTHOR
| Gregory V. Richardson (omomom(AT)hotmail.com), Oct 14 2002
|
|
|
EXTENSIONS
| Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
|
| |
|
|
