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

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

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

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.

R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (32).

Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.

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).

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.

Cf. A000931, A001608.

Sequence in context: A078027 A134816 A228361 * A290697 A290821 A072493

Adjacent sequences: A182094 A182095 A182096 * A182098 A182099 A182100

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Apr 11 2012

Status

approved