|
| |
|
|
A001081
|
|
a(n) = 16a(n-1) - a(n-2).
(Formerly M4573 N1949)
|
|
10
| |
|
|
1, 8, 127, 2024, 32257, 514088, 8193151, 130576328, 2081028097, 33165873224, 528572943487, 8424001222568, 134255446617601, 2139663144659048, 34100354867927167, 543466014742175624, 8661355881006882817
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Chebyshev's polynomials T(n,x) evaluated at x=8.
The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 63*b(n)^2 = +1 with b(n)= A077412(n-1), n>=1 and b(0)=0.
Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(7) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 14 2004
a(7+14k)-1 and a(7+14k)+1 are consecutive odd powerful numbers. The first pair is 130576328+-1. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006
|
|
|
REFERENCES
| H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.
M. Davis, One equation to rule them all, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773, http://www.rand.org/pubs/research_memoranda/RM5494.html
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 281.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
|
|
|
FORMULA
| For all members x of the sequence, 7*x^2 - 7 is a square. Lim. n-> Inf. a(n)/a(n-1) = 8 + 3*Sqrt(7). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002
a(n) = T(n, 8) = (S(n, 16)-S(n-2, 16))/2, with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 16)= A077412(n).
a(n) = ((8+3*sqrt(7))^n + (8-3*sqrt(7))^n)/2.
a(n) = sqrt(63*A077412(n-1)^2 + 1), n>=1, (cf. Richardson comment).
G.f.: (1-8*x)/(1-16*x+x^2).
a(n) = 16*a(n-1) - a(n-2) with a(1)=1 and a(2)=8. - Sture Sjöstedt, Nov 18 2011
|
|
|
MAPLE
| A001081:=-(-1+8*z)/(1-16*z+z**2); [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
| LinearRecurrence[{16, -1}, {1, 8}, 50]
|
|
|
PROG
| sage: [lucas_number2(n, 16, 1)/2 for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
|
|
|
CROSSREFS
| Cf. A090727.
Cf. A001080, A010727 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 16 2009]
Sequence in context: A035130 A055762 A029472 * A178790 A034220 A034239
Adjacent sequences: A001078 A001079 A001080 * A001082 A001083 A001084
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
|
| |
|
|
