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A077846
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Expansion of 1/(1-3*x+2*x^3).
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2
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1, 3, 9, 25, 69, 189, 517, 1413, 3861, 10549, 28821, 78741, 215125, 587733, 1605717, 4386901, 11985237, 32744277, 89459029, 244406613, 667731285, 1824275797, 4984014165, 13616579925, 37201188181, 101635536213, 277673448789
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 17 2004
A Whitney transform of 2^n (see Benoit Cloitre formula and A004070). The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry (pbarry(AT)wit.ie), Feb 16 2005
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LINKS
| Fan Chung, Ron Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194
Index to sequences with linear recurrences with constant coefficients, signature (3,0,-2)
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FORMULA
| a(n) = 3*a(n-1)-2*a(n-3) = 2*A057960(n)-1 = round[2*A028859(n)/sqrt(3)-1/3] = sum_i(b(n, i)) where b(n, 0)=b(n, 6)=0, b(0, 3)=1, b(0, i)=0 if i<>3 and b(n+1, i)=b(n, i-1)+b(n, i)+b(n, i+1) if 0<i<6 [i.e. the number of three-choice paths along a corridor width 5, starting from the middle]. - Henry Bottomley (se16(AT)btinternet.com), Mar 06 2003
Binomial transform of A068911. a(n)=(1+sqrt(3))^n(2+sqrt(3))/3+(1-sqrt(3))^n(2-sqrt(3))/3-1/3 - Paul Barry (pbarry(AT)wit.ie), Feb 17 2004
a(0)=1, for n>=1 a(n)=ceil((1+sqrt(3))*a(n-1)). - Benoit Cloitre, Jun 19 2004.
a(n)=sum(i=0, n, sum(j=0, n, 2^j*binomial(j, i-j))) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 23 2004
a(n) = 2(a(n-1) +a(n-2))+1, n>1 [From Gary Detlefs (gdetlefs(AT)aol.com), Jun 20 2010]
a(n) = (2*A052945(n+1)-1)/3. - R. J. Mathar, Mar 31 2011
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MATHEMATICA
| a=0; b=0; lst={a, b}; Do[c=2*(a+b)+1; AppendTo[lst, c]; a=b; b=c, {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 01 2010]
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PROG
| (PARI) a(n)=sum(i=0, n, sum(j=0, n, 2^j*binomial(j, i-j)))
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CROSSREFS
| First differences are in A002605.
Sequence in context: A094292 A201533 A000242 * A005322 A103780 A206727
Adjacent sequences: A077843 A077844 A077845 * A077847 A077848 A077849
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002
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EXTENSIONS
| Name changed by Arkadiusz Wesolowski, Dec 06 2011
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