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A182097
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Expansion of 1/(1-x^2-x^3).
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28
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1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081, 922111, 1221537, 1618192, 2143648, 2839729, 3761840, 4983377, 6601569, 8745217
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OFFSET
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0,6
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COMMENTS
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Number of compositions (ordered partitions) into parts 2 and 3. - Joerg Arndt, Aug 21 2013
a(n) is the top left entry of the n-th power of any of the 3X3 matrices [0, 1, 1; 0, 0, 1; 1, 0, 0], [0, 1, 0; 1, 0, 1; 1, 0, 0], [0, 1, 1; 1, 0, 0; 0, 1, 0] or [0, 0, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
Conjectured values of d(n), the dimension of a Z-module in MZV(conv). See the Waldschmidt link. - Michael Somos, Mar 14 2014
Shannon et al. (2006) call these the Van der Laan numbers. - N. J. A. Sloane, Jan 11 2022
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REFERENCES
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A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See R_n.
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
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Michel Waldschmidt, Multiple Zeta values and Euler-Zagier numbers, Slides, 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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FORMULA
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G.f.: 1 / (1 - x^2 - x^3).
a(n) = a(n-2) + a(n-3) for all n in Z.
a(n+4) = A164001(n) if n > 1. - (End)
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EXAMPLE
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G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
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MATHEMATICA
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a[ n_] := If[n < 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, n}]]; (* Michael Somos, Dec 13 2013 *)
CoefficientList[Series[1/(1-x^2-x^3), {x, 0, 60}], x] (* or *) LinearRecurrence[ {0, 1, 1}, {1, 0, 1}, 70] (* Harvey P. Dale, Dec 04 2014 *)
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PROG
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(PARI) {a(n) = if( n<0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n))}; /* Michael Somos, Dec 13 2013 */
(PARI) Vec(1/(1-x^2-x^3) + O(x^99)) \\ Altug Alkan, Sep 02 2016
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3))); // G. C. Greubel, Aug 11 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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