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A071618
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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).
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4
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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LINKS
| S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ...
Harvey P. Dale, Table of n, a(n) for n = 0..1000
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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
a(0)=0, a(1)=1, a(2)=3, a(3)=8, a(4)=23, a(n)=3*a(n-1)-a(n-2)+ a(n-3)- 3*a(n-4)+a(n-5) [From Harvey P. Dale, Dec 18 2011]
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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} ]
LinearRecurrence[{3, -1, 1, -3, 1}, {0, 1, 3, 8, 23}, 30] (* or *) CoefficientList[ Series[x (1+x^3)/((1-x^3)*(1-3x+x^2)), {x, 0, 30}], x] (* From Harvey P. Dale, Dec 18 2011 *)
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PROG
| (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 */
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CROSSREFS
| Sequence in context: A027236 A027244 A108457 * A146998 A103819 A147484
Adjacent sequences: A071615 A071616 A071617 * A071619 A071620 A071621
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 21 2002
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EXTENSIONS
| More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 23 2002
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