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.

A114647 Expansion of (3 -4*x -3*x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

3, 2, 7, 12, 31, 70, 171, 408, 987, 2378, 5743, 13860, 33463, 80782, 195027, 470832, 1136691, 2744210, 6625111, 15994428, 38613967, 93222358, 225058683, 543339720, 1311738123, 3166815962, 7645370047, 18457556052, 44560482151, 107578520350, 259717522851
Offset: 0

Views

Author

Creighton Dement, Feb 18 2006

Keywords

Comments

Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

Links

Crossrefs

Cf. A000129, A059841, A100828, A114688, A114689, A114695, A114696, A114697.

Programs

  • Magma
    I:=[3,2,7,12]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 24 2021
  • Mathematica
    Table[Fibonacci[n+1, 2] +1+(-1)^n, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)
  • PARI
    Vec((3-4*x-3*x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, May 26 2016
  • Sage
    [lucas_number1(n+1,2,-1) +(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021

Formula

G.f.: (3 -4*x -3*x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(n) = A000129(n+1) + 2*A059841(n). - R. J. Mathar, Nov 10 2009
From Colin Barker, May 26 2016: (Start)
a(n) = 1 + (-1)^n + ((1+sqrt(2))^(1+n) - (1-sqrt(2))^(1+n))/(2*sqrt(2)).
a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) for n>3. (End)
a(n) = A000129(n+1) + 1 + (-1)^n. - G. C. Greubel, May 24 2021

A114688 Expansion of (1 +3*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 5, 11, 30, 71, 175, 421, 1020, 2461, 5945, 14351, 34650, 83651, 201955, 487561, 1177080, 2841721, 6860525, 16562771, 39986070, 96534911, 233055895, 562646701, 1358349300, 3279345301, 7917039905, 19113425111, 46143890130, 111401205371, 268946300875
Offset: 0

Views

Author

Creighton Dement, Feb 18 2006

Keywords

Comments

Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

Links

Crossrefs

Cf. A000129, A100828, A114647, A114689, A114695, A114696, A114697.

Programs

  • Magma
    I:=[1,5,11,30]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 24 2021
  • Maple
    Pell:= proc(n) option remember;
    if n<2 then n
  else 2*Pell(n-1) + Pell(n-2)
    fi; end:
seq((10*Pell(n+1) -3*(1+(-1)^n))/4, n=0..40); # G. C. Greubel, May 24 2021
  • Mathematica
    CoefficientList[Series[(-1-3x+x^2)/((1-x)(x+1)(x^2+2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-2,-1},{1,5,11,30},40] (* Harvey P. Dale, Dec 18 2012 *)
  • PARI
    Vec((-1-3*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, May 26 2016
  • Sage
    [(10*lucas_number1(n+1,2,-1) -3*(1+(-1)^n))/4 for n in (0..30)] # G. C. Greubel, May 24 2021

Formula

G.f.: (1 +3*x -x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(0)=1, a(1)=5, a(2)=11, a(3)=30, a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4). - Harvey P. Dale, Dec 18 2012
a(n) = (-6 - 6*(-1)^n + 5*sqrt(2)*( (1+sqrt(2))^(1+n) - (1-sqrt(2))^(1+n) ))/8. - Colin Barker, May 26 2016
a(n) = (10*A000129(n+1) - 3*(1 + (-1)^n))/4. - G. C. Greubel, May 24 2021

A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 3, 9, 22, 55, 133, 323, 780, 1885, 4551, 10989, 26530, 64051, 154633, 373319, 901272, 2175865, 5253003, 12681873, 30616750, 73915375, 178447501, 430810379, 1040068260, 2510946901, 6061962063, 14634871029, 35331704122, 85298279275, 205928262673
Offset: 0

Views

Author

Creighton Dement, Feb 18 2006

Keywords

Comments

Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').

Links

Crossrefs

Cf. A000129, A002203, A005409, A100828, A111954, A113224.
Cf. A114647, A114688, A114689, A114695, A116699.

Programs

  • Mathematica
    Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n,0,30}] (* G. C. Greubel, May 24 2021 *)
  • PARI
    Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 24 2015
  • Sage
    [(4*lucas_number1(n+2,2,-1) -2*lucas_number1(n+1,2,-1) -3 +(-1)^n)/4 for n in (0..30)] # G. C. Greubel, May 24 2021

Formula

a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).
G.f.: (1+x+x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
From Raphie Frank, Oct 01 2012: (Start)
a(2*n) = A216134(2*n+1).
a(2*n+1) = A006452(2*n+3)-1.
Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)
a(n) = (2*A078343(n+2) - A010694(n))/4. - R. J. Mathar, Oct 02 2012
From Colin Barker, May 26 2016: (Start)
a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - G. C. Greubel, May 24 2021

A114696 Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 6, 15, 40, 97, 238, 575, 1392, 3361, 8118, 19599, 47320, 114241, 275806, 665855, 1607520, 3880897, 9369318, 22619535, 54608392, 131836321, 318281038, 768398399, 1855077840, 4478554081, 10812186006, 26102926095, 63018038200, 152139002497, 367296043198
Offset: 0

Views

Author

Creighton Dement, Feb 18 2006

Keywords

Comments

Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.
Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

Links

Crossrefs

Cf. A000129, A002203, A005409, A100828, A111954, A113224.
Cf. A114647, A114688, A114689, A114695, A114697.

Programs

  • Maple
    Q:= proc(n) option remember; # Q=A002203
    if n<2 then 2
  else 2*Q(n-1) + Q(n-2)
    fi; end:
seq((Q(n+2) -3 -(-1)^n)/2, n=0..40); # G. C. Greubel, May 24 2021
  • Mathematica
    CoefficientList[Series[(1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)), {x,0,30}], x] (* or *) LinearRecurrence[{2,2,-2,-1}, {1,6,15,40}, 30] (* Harvey P. Dale, Jan 23 2014 *)
  • PARI
    Vec((1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^30)) \\ Colin Barker, May 26 2016
  • Sage
    [(lucas_number2(n+2,2,-1) -3 -(-1)^n)/2 for n in (0..30)] # G. C. Greubel, May 24 2021

Formula

G.f.: (1 +4*x +x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Harvey P. Dale, Jan 23 2014
a(n) = (-3 - (-1)^n + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2)))/2. - Colin Barker, May 26 2016
From G. C. Greubel, May 24 2021: (Start)
a(n) = 3*A000129(n+1) + A000129(n) - (3 + (-1)^n)/2.
a(n) = (1/2)*(A002203(n+2) - 3 - (-1)^n). (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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