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A182097 Expansion of 1/(1-x^2-x^3). 23
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

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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).

Index entries for linear recurrences with constant coefficients, signature (0,1,1)

FORMULA

G.f.: 1 / (1 - x^2 - x^3).

a(n) = A000931(n+3).

a(n) = A176971(-n). a(n) = a(n-2) + a(n-3) for all n in Z. - Michael Somos, Dec 13 2013

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. - Michael Somos, Dec 13 2013

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

CROSSREFS

The following are basically all variants of the same sequence: A000931, 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 A072493 A064324

Adjacent sequences:  A182094 A182095 A182096 * A182098 A182099 A182100

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Apr 11 2012

STATUS

approved

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Last modified April 27 01:03 EDT 2018. Contains 303149 sequences. (Running on oeis4.)