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A182097 Expansion of 1/(1-x^2-x^3). 28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
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
REFERENCES
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).
LINKS
M. Hoffman, The algebra of multiharmonic series, Journ. of Alg., Vol. 192, Issue 2 (Aug 1997), 477-495.
I. E. Leonard and A. C. F. Liu, A familiar recurrence occurs again, Amer. Math. Monthly, 119 (2012), 333-336.
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).
FORMULA
G.f.: 1 / (1 - x^2 - x^3).
a(n) = A000931(n+3).
From Michael Somos, Dec 13 2013: (Start)
a(n) = A176971(-n).
a(n) = a(n-2) + a(n-3) for all n in Z.
a(n-7) = A133034(n).
a(n-5) = A078027(n).
a(n-3) = A000931(n).
a(n+2) = A134816(n).
a(n+4) = A164001(n) if n > 1. - (End)
a(n) = (A001608(n) - A000931(n))/2. - Elmo R. Oliveira, Dec 31 2022
EXAMPLE
G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
The following are basically all variants of the same sequence: A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.
Sequence in context: A078027 A134816 A228361 * A290697 A290821 A072493
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Apr 11 2012
STATUS
approved

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Last modified March 19 03:27 EDT 2024. Contains 370952 sequences. (Running on oeis4.)