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.

Showing 1-4 of 4 results.

A111641 Expansion of -(1+x+3*x^2+x^3)/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

1, -5, 25, -107, 433, -1697, 6529, -24839, 93841, -352973, 1323961, -4957139, 18539041, -69282185, 258790465, -966364367, 3607837153, -13467809237, 50270219929, -187629535739, 700287673681, -2613617125553, 9754412512321, -36404592257879, 135865306871281
Offset: 0

Views

Author

Creighton Dement, Aug 10 2005

Keywords

Comments

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].
Floretion Algebra Multiplication Program, FAMP Code: 1lestesseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Links

Crossrefs

Cf. A111639, A111640, A111642, A111643, A111644, A000126.

Programs

  • Mathematica
    CoefficientList[Series[-(1+x+3x^2+x^3)/((x^2+4x+1)(x^2-2x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{-6,-8,2,1},{1,-5,25,-107},30] (* Harvey P. Dale, Oct 12 2017 *)
  • PARI
    Vec((1 + x + 3*x^2 + x^3) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, Apr 29 2019

Formula

a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. - Colin Barker, Apr 29 2019

A111642 Expansion of 2*(x-1)*(x+1)/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

2, -12, 54, -224, 890, -3452, 13198, -50016, 188498, -707916, 2652678, -9925760, 37105802, -138631292, 517742494, -1933118784, 7216615970, -26937891852, 100545928278, -375272321696, 1400607336218, -5227311479036, 19509011469358, -72809634633120, 271731700422002
Offset: 0

Views

Author

Creighton Dement, Aug 10 2005

Keywords

Comments

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].
Floretion Algebra Multiplication Program, FAMP Code: 1ibaseseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Links

Crossrefs

Cf. A111639, A111640, A111641, A111643, A111644, A000126.

Programs

  • PARI
    Vec(2*(1 - x)*(1 + x) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, Apr 29 2019

Formula

a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. - Colin Barker, Apr 29 2019

A111643 Expansion of 2*(x+1)^2/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

-2, 8, -34, 136, -530, 2032, -7714, 29104, -109378, 410040, -1534722, 5738360, -21441682, 80083808, -299027394, 1116348896, -4167148290, 15554127592, -58053908834, 216672484584, -808662529938, 3018041612880, -11263658377442, 42036964786320, -156885101002562
Offset: 0

Views

Author

Creighton Dement, Aug 10 2005

Keywords

Comments

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].
Floretion Algebra Multiplication Program, FAMP Code: 1baseiseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Links

Crossrefs

Cf. A111639, A111640, A111641, A111642, A111644, A000126.

Programs

  • PARI
    Vec(-2*(1 + x)^2 / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^40)) \\ Colin Barker, May 01 2019

Formula

From Colin Barker, May 01 2019: (Start)
a(n) = (-3*(-1-sqrt(2))^(1+n) - 3*(-1+sqrt(2))^(1+n) - 9*(-2-sqrt(3))^n - 5*sqrt(3)*(-2-sqrt(3))^n - 9*(-2+sqrt(3))^n + 5*sqrt(3)*(-2+sqrt(3))^n) / 6.
a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3.
(End)

A111645 Expansion of (x+1)*(1-3*x)/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

-1, 8, -37, 156, -625, 2436, -9341, 35464, -133809, 502896, -1885317, 7056580, -26384961, 98589388, -368228797, 1374944336, -5133041825, 19160828056, -71518973861, 266936079404, -996276071249, 3718290672596, -13877182280637, 51791152239960, -193289149920721
Offset: 0

Views

Author

Creighton Dement, Aug 10 2005

Keywords

Comments

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].
Floretion Algebra Multiplication Program, FAMP Code: 1jbasejseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Links

Crossrefs

Cf. A111639, A111640, A111641, A111642, A111643, A111644, A000126.

Programs

  • Mathematica
    CoefficientList[Series[(x+1)(1-3x)/((x^2+4x+1)(x^2-2x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{-6,-8,2,1},{-1,8,-37,156},30] (* Harvey P. Dale, Nov 19 2015 *)

Formula

a(0)=-1, a(1)=8, a(2)=-37, a(3)=156, a(n)=-6*a(n-1)-8*a(n-2)+2*a(n-3)+a(n-4). - Harvey P. Dale, Nov 19 2015
2*a(n) = -7*A125905(n)-A125905(n-1) -A077985(n-1)+5*A077985(n). - R. J. Mathar, Sep 11 2019
Showing 1-4 of 4 results.

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