|
| |
|
|
A057089
|
|
Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
|
|
10
| |
|
|
1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.
|
|
|
REFERENCES
| A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=6.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(39) and (45),rhs, m=6.
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
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*(a(n-1)+6*a(n-2)), a(0)=1, a(1)=6
a(n)= S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-6*x-6*x^2).
a(n)=Sum_{k, 0<=k<=n}5^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
a(n)=-(1/30)*sqrt(15)*[3-sqrt(15)]^(n+1)+(1/30)*sqrt(15)*[3+sqrt(15)]^(n+1), with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 20 2008]
|
|
|
MATHEMATICA
| Join[{a=0, b=1}, Table[c=6*b+6*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)
LinearRecurrence[{6, 6}, {1, 6}, 40] (* From Harvey P. Dale, Nov 05 2011 *)
|
|
|
PROG
| (Other) sage: [lucas_number1(n, 6, -6) for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
(MAGMA) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)+6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
|
|
|
CROSSREFS
| Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.
Sequence in context: A062310 A105482 A157335 * A110711 A156361 A055272
Adjacent sequences: A057086 A057087 A057088 * A057090 A057091 A057092
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000
|
|
|
EXTENSIONS
| First formula corrected by Harvey P. Dale, Nov 05 2011
|
| |
|
|
