cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A073496 Expansion of (3 + 2*x + 3*x^2)/(1 + x + 3*x^2 - x^3).

Original entry on oeis.org

3, -1, -5, 11, 3, -41, 43, 83, -253, 47, 795, -1189, -1149, 5511, -3253, -14429, 29699, 10335, -113861, 112555, 239363, -690889, 85355, 2226675, -3173629, -3421041, 15168603, -8079109, -40847741, 80253671, 34210443, -315819197, 293441539, 688226495, -1884370309, 113132363, 6228205059
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 03 2002

Keywords

Comments

Old name was "a(2n), where a(n) is A073145".

Links

Crossrefs

Bisection of A073145.

Programs

  • Mathematica
    CoefficientList[Series[(3 + 2*x + 3*x^2)/(1 + x + 3*x^2 - x^3), {x, 0, 50}], x]
LinearRecurrence[{-1,-3,1},{3,-1,-5},40] (* Harvey P. Dale, Aug 22 2018 *)

Formula

G.f.: (3 + 2*x + 3*x^2)/(1 + x + 3*x^2 - x^3).
a(2n)=-a(2n-2)-3a(2n-4)+a(2n-6), a(0)=3, a(2)=-1, a(4)=-5.
Recurrence: a(n) = a(n-3) - 3a(n-2) - a(n-1), starting 3,-1,-5.

A073748 a(n) = S(n)*S(n-1), where S(n) are the generalized tribonacci numbers A001644.

Original entry on oeis.org

-3, 3, 3, 21, 77, 231, 819, 2769, 9301, 31571, 106763, 361045, 1221685, 4132743, 13980747, 47297217, 160004685, 541291715, 1831178355, 6194830005, 20956959933, 70896891079, 239842458947, 811381229009, 2744883043045, 9285872805715, 31413882695739, 106272403946805
Offset: 0

Views

Author

Maio Catalani (mario.catalani(AT)unito.it), Aug 08 2002

Keywords

Links

Crossrefs

Cf. A001644, A073145.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (-3+9*x+6*x^2+24*x^3+5*x^4-x^5)/(1-2*x-3*x^2-6*x^3 +x^4+x^6) )); // G. C. Greubel, Apr 21 2019
  • Mathematica
    CoefficientList[Series[(-3+9*x+6*x^2+24*x^3+5*x^4-x^5)/(1-2*x-3*x^2-6*x^3 +x^4+x^6), {x, 0, 30}], x]
Join[{-3},Times@@@Partition[LinearRecurrence[{1,1,1},{3,1,3},30],2,1]] (* or *) LinearRecurrence[{2,3,6,-1,0,-1},{-3,3,3,21,77,231},30] (* Harvey P. Dale, Nov 18 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec((-3+9*x+6*x^2+24*x^3+5*x^4-x^5)/(1-2*x-3*x^2-6*x^3 +x^4+x^6)) \\ G. C. Greubel, Apr 21 2019
  • Sage
    ((-3+9*x+6*x^2+24*x^3+5*x^4-x^5)/(1-2*x-3*x^2-6*x^3 +x^4+x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 21 2019

Formula

a(n) = S(2*n-1) + C(n-1) - C(n-2), where S(n) is A001644, C(n) is A073145.
G.f.: (-3 + 9*x + 6*x^2 + 24*x^3 + 5*x^4 - x^5)/(1 - 2*x - 3*x^2 - 6*x^3 + x^4 + x^6).
a(0)=-3, a(1)=3, a(2)=3, a(3)=21, a(4)=77, a(5)=231, a(n) = 2*a(n-1) + 3*a(n-2) + 6*a(n-3) - a(n-4) - a(n-6). - Harvey P. Dale, Nov 18 2013
Previous Showing 11-12 of 12 results.

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