|
| |
|
|
A014551
|
|
Jacobsthal-Lucas numbers.
|
|
35
| |
|
|
2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 2147483647, 4294967297, 8589934591
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
COMMENTS
| Also gives the number of points of period n in the subshift of finite type corresponding to the square matrix A=[1,2;1,0] (this is then given by trace(A^n)). - Thomas Ward (t.ward(AT)uea.ac.uk), Mar 07 2001
Sequence is identical to its signed inverse binomial transform. - Paul Curtz (bpcrtz(AT)free.fr), Jul 11 2008
a(n) can be expressed in terms of values of the Fibonacci polynomials F_n(x), computed at x=1/sqrt(2). [From Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008]
|
|
|
REFERENCES
| G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.
Horadam, A. F. ``Jacobsthal and Pell Curves.'' Fib. Quart. 26, 79-83, 1988.
Horadam, A. F. ``Jacobsthal Representation Numbers.'' Fib Quart. 34, 40-54, 1996.
Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type)
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,2).
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Jacobsthal Number
Index entries for sequences related to Chebyshev polynomials.
T. Amdeberhan, A note on Fibonacci-type polynomials [From Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008]
|
|
|
FORMULA
| a(n+1) = 2 * a(n) - (-1)^n * 3.
a(n) = 2^n + (-1)^n. G.f.: (2-x)/(1-x-2*x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 07 2001
E.g.f.: exp(x)+exp(-2x) produces a signed version. - Paul Barry (pbarry(AT)wit.ie), Apr 27 2003
a(n+1)=Sum{k=0..floor(n/2), binomial(n-1, 2k)3^(2k)/2^(n-2)}. - Paul Barry (pbarry(AT)wit.ie), Feb 21 2003
0, 1, 5, 7 ... is 2^n-2*0^n+(-1)^n, the 2nd inverse binomial transform of (2^n-1)^2 (A060867) - Paul Barry (pbarry(AT)wit.ie), Sep 05 2003
a(n)=2T(n, i/(2sqrt(2)))(-i*sqrt(2))^n with i^2=-1 - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
a(n)=(A078008(n)+A001045(n+1)) - Paul Barry (pbarry(AT)wit.ie), Feb 12 2004
a(n)=2*A001045(n+1)-A001045(n) - Paul Barry (pbarry(AT)wit.ie), Mar 22 2004
a(0)=2, a(1)=1, a(n)=a(n-1)+2*a(n-2) for n>1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 07 2006
a(2n+1)=Product(d divides 2n+1, cyclotomic(d,2)). a(2^k*(2n+1))=Product(d divides 2n+1,cyclotomic(2d,2^(2^k))) - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 12 2007
a(n)=2^{(n-1)/2}F_{n-1}(1/sqrt(2))+2^{(n+2)/2}F_{n-2}(1/sqrt(2)). [From Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008]
|
|
|
MATHEMATICA
| f[n_]:=2/(n+1); x=4; Table[x=f[x]; Denominator[x], {n, 0, 5!}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 12 2010]
|
|
|
PROG
| (Other) sage: [lucas_number2(n, 1, -2) for n in xrange(0, 32)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
|
|
|
CROSSREFS
| Cf. A001045 A019322 A066845.
Sequence in context: A105459 A183946 A005297 * A175002 A088014 A193662
Adjacent sequences: A014548 A014549 A014550 * A014552 A014553 A014554
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
|
|
|
EXTENSIONS
| More terms from Patrick De Geest (pdg(AT)worldofnumbers.com), Jun 15 1998.
|
| |
|
|
