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A079398
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Conjectured values of d(n), the dimension of a Z-module in MZV(conv).
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20
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0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 7, 8, 9, 12, 15, 17, 21, 27, 32, 38, 48, 59, 70, 86, 107, 129, 156, 193, 236, 285, 349, 429, 521, 634, 778, 950, 1155, 1412, 1728, 2105, 2567, 3140, 3833, 4672, 5707, 6973, 8505, 10379, 12680, 15478, 18884, 23059, 28158, 34362
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| From the conjecture of Zagier, Drinfeld, Kontsevich and Goncharov (see link).
P(0)=P(1)=P(2)=P(3)=1, for m>3: P(m) = P(m-3) + P(m-4) is the 3rd sequence in the series: Fibonacci sequence, Padovan sequence, ... The Padovan sequence (whose ratio of successive terms approaches the plastic constant) is similar to the Perrin sequence. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 23 2005
Binomial transform yields A079398 without the initial (0,1,1,1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 09 2008
a(n+1) corresponds to the diagonal sums of "triangle" : 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,4,6,4,1 ; ..., rows of Pascal's triangle (A007318) repeated three times . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 13 2008]
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REFERENCES
| Michel Waldschmidt, "Multiple Zeta values and Euler-Zagier numbers", in Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).
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LINKS
| Michel Waldschmidt, Multiple Zeta values and Euler-Zagier numbers
Eric Weisstein's World of Mathematics, Padovan Sequence.
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FORMULA
| a(1)=0, a(2)=a(3)=a(4)=1; for n>=4, a(n)=a(n-2)+a(n-3).
a(n)=sum{k=0..floor((n-1)/2), binomial(floor((n-k-1)/3), k)} (offset 0); a(n)=sum{k=0..floor(n/2), binomial(floor((n-k-1)/3), k)}-0^n. (offset 0). - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004
For n>1, a(n) = P(n-2) where P(n) is defined by: P(0)=P(1)=P(2)=P(3)=1, for m>3: P(m) = P(m-3) + P(m-4). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 23 2005
The same sequence may be constructed as follows: Let M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}}; v[1] = {1, 1, 1, 1}; v[n] = M.v[n - 1]. Then a(n) = v[n][[1]]. - Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2006
O.g.f.: -x^2*(1+x+x^2)/(-1+x^3+x^4). a(n)=A017817(n-1)+A017817(n-2)+A017817(n-3). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 09 2008
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MATHEMATICA
| a=b=c=1; d=1; lst={0, a, b, c, d}; Do[AppendTo[lst, e=a+b]; a=b; b=c; c=d; d=e, {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 28 2010]
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CROSSREFS
| Cf. A000931.
Sequence in context: A077564 A088044 A029051 * A071988 A029050 A066920
Adjacent sequences: A079395 A079396 A079397 * A079399 A079400 A079401
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 16 2003
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