|
| |
|
|
A004190
|
|
Expansion of 1/(1-11*x+x^2).
|
|
12
| |
|
|
1, 11, 120, 1309, 14279, 155760, 1699081, 18534131, 202176360, 2205405829, 24057287759, 262424759520, 2862615066961, 31226340977051, 340627135680600, 3715672151509549, 40531766530924439, 442133759688659280
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Chebyshev or generalized Fibonacci sequence.
This is the m=13 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..12 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913 and A004189. The m=1..3 (signed) sequences are A049347, A056594, A010892.
All positive integer solutions of Pell equation b(n)^2 - 117*a(n)^2 = +4 together with b(n+1)=A057076(n+1), n>=0. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 11's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
|
|
|
REFERENCES
| A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=11, q=-1.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=13.
|
|
|
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
| Recursion: a(n)=11*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n)=S(2*n+1, sqrt(13))/sqrt(13) = S(n, 11); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
G.f.: 1/(1-11*x+x^2).
a(n)=((11+3*sqrt(13))^(n+1)-(11-3*sqrt(13))^(n+1))/(2^(n+1)*3*sqrt(13)). [Rolf Pleisch (r_pleisch(AT)gmx.ch), May 22 2011]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*10^k. - DELEHAM Philippe, Feb 10 2012
|
|
|
MAPLE
| with (combinat):seq(fibonacci(2*n, 3)/3, n=1..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008
|
|
|
MATHEMATICA
| Join[{a=1, b=11}, Table[c=11*b-a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 20 2011*)
|
|
|
PROG
| sage: [lucas_number1(n, 11, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
|
|
|
CROSSREFS
| A049310, A004189. a(n)=sqrt((A057076(n+1)^2 - 4)/117).
Sequence in context: A060498 A171316 A081122 * A089707 A084969 A045592
Adjacent sequences: A004187 A004188 A004189 * A004191 A004192 A004193
|
|
|
KEYWORD
| nonn,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002
|
| |
|
|
