|
Search: id:A071618
|
|
|
| A071618 |
|
a(n+1) - 3*a(n) + a(n-1) = (2/3)(1+w^(n+1)+w^(2n+2)), where w = exp(2 Pi I / 3). |
|
+0
4
|
|
| 0, 1, 3, 8, 23, 61, 160, 421, 1103, 2888, 7563, 19801, 51840, 135721, 355323, 930248, 2435423, 6376021, 16692640, 43701901, 114413063, 299537288, 784198803, 2053059121, 5374978560, 14071876561, 36840651123, 96450076808, 252509579303
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
The sequence is closely related to the third term in the continued fraction expansion of 2(F(4n)+F(2n))/phi where F is the Fibonacci sequence. For any k smaller than a(n), k*F(2n)*phi has to be rounded by excess, for any k greater than a(n), k*F(2n)*phi has to be rounded by default. - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Aug 31 2004
|
|
LINKS
|
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ...
|
|
FORMULA
|
a(n) = floor ( phi^2n / 2 ) = floor ( (Lucas(2n)-1) / 2 ). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Aug 31 2004
a(-n)= a(n) . a(n)= 2*a(n-1) +a(n-2) +2*a(n-3) -a(n-4) +2 . - Michael Somos Mar 08 2007
G.f.: x*(1+x^3)/ ((1-x^3)* (1-3*x+x^2)) . - Michael Somos Mar 08 2007
|
|
MATHEMATICA
|
a[ -1 ] = 0; a[ 0 ] = 1; w = Exp[ 2Pi*I/3 ]; a[ n_ ] := a[ n ] = Simplify[ (2/3)(1 + w^n + w^(2n)) + 3a[ n - 1 ] - a[ n - 2 ] ]; Table[ a[ n ], {n, -1, 28} ]
|
|
PROGRAM
|
(PARI) u=0; v=1; for(n=1, 30, print1(a=3*v-u+2*!(n%3), " "); u=v; v=a) (from Thomas Baruchel)
(PARI) {a(n)= ( fibonacci(2*n+1)+ fibonacci(2*n-1)+ (n%3>0))/2- 1 } /* Michael Somos Mar 08 2007 */
(PARI) {a(n)= n=abs(n); polcoeff( x*(1+x^3)/ ((1-x^3)* (1-3*x+x^2)) +x*O(x^n), n)} /* Michael Somos Mar 08 2007 */
|
|
CROSSREFS
|
Sequence in context: A027236 A027244 A108457 this_sequence A146998 A103819 A147484
Adjacent sequences: A071615 A071616 A071617 this_sequence A071619 A071620 A071621
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com), Jun 21 2002
|
|
EXTENSIONS
|
More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 23 2002
|
|
|
Search completed in 0.002 seconds
|
