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 1-6 of 6 results.

A111639 Expansion of (3+8*x-3*x^2-2*x^3)/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

-3, 10, -33, 114, -403, 1450, -5281, 19394, -71619, 265450, -986241, 3670002, -13670803, 50957770, -190026433, 708824834, -2644492803, 9867263050, -36820012641, 137401810674, -512760729619, 1913577130090, -7141393334881, 26651623320002, -99464199710403
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: 2tesseq[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. A111640, A111641, A111642, A111643, A000126.

Programs

  • Mathematica
    LinearRecurrence[{-6,-8,2,1},{-3,10,-33,114},30] (* Harvey P. Dale, Jul 04 2019 *)
  • PARI
    Vec(-(3 + 8*x - 3*x^2 - 2*x^3) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, May 11 2019

Formula

From Colin Barker, May 11 2019: (Start)
a(n) = ((-1-sqrt(2))^(1+n) + (-1+sqrt(2))^(1+n) - 2*(-2-sqrt(3))^n - sqrt(3)*(-2-sqrt(3))^n - 2*(-2+sqrt(3))^n + sqrt(3)*(-2+sqrt(3))^n) / 2.
a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. (End)

A111640 Expansion of (-1+3*x+x^2-x^3)/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

1, -9, 45, -195, 793, -3117, 12013, -45751, 172961, -650849, 2441917, -9144539, 34203161, -127829669, 477505565, -1783134255, 6657304833, -24851573497, 92762239373, -346229372851, 1292232479961, -4822886991709, 17999765604237, -67177262104679, 250711906290721
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: 1vesseq[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, A111641, A111642, A111643, A000126.

Programs

  • PARI
    Vec((1 - 3*x - 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

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

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)

A111644 Expansion of -(1+x^2)/((x^2+4*x+1)*(x^2-2*x-1)).

Original entry on oeis.org

1, -6, 29, -124, 501, -1962, 7545, -28696, 108393, -407662, 1528981, -5724500, 21408221, -80003026, 298832369, -1115878064, 4166011601, -15551383382, 58047283725, -216656490156, 808623915973, -3017948390522, 11263433318761, -42036421446600, 156883789264441
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: 1basejseq[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, A111645, A000126.

Programs

  • Mathematica
    CoefficientList[Series[-(1+x^2)/((x^2+4x+1)(x^2-2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{-6,-8,2,1},{1,-6,29,-124},40] (* Harvey P. Dale, May 23 2015 *)

Formula

a(0)=1, a(1)=-6, a(2)=29, a(3)=-124, a(n)=-6*a(n-1)-8*a(n-2)+ 2*a(n-3)+ a(n-4). - Harvey P. Dale, May 23 2015

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 1-6 of 6 results.

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