A114696 -id:A114696 - cp's OEIS frontend

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

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

Creighton Dement, Feb 18 2006

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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

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

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

Table[Fibonacci[n+1, 2] +1+(-1)^n, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)

Vec((3-4*x-3*x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, May 26 2016

[lucas_number1(n+1,2,-1) +(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021

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 +3x -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

Creighton Dement, Feb 18 2006

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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

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

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

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

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 *)

Vec((-1-3*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, May 26 2016

[(10*lucas_number1(n+1,2,-1) -3*(1+(-1)^n))/4 for n in (0..30)] # G. C. Greubel, May 24 2021

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

A114689 Expansion of (1 +4x -x^2)/((1-x^2)(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 6, 13, 36, 85, 210, 505, 1224, 2953, 7134, 17221, 41580, 100381, 242346, 585073, 1412496, 3410065, 8232630, 19875325, 47983284, 115841893, 279667074, 675176041, 1630019160, 3935214361, 9500447886, 22936110133, 55372668156, 133681446445, 322735561050

Offset: 0

Creighton Dement, Feb 18 2006

Elements of odd index give match to A075848: 2*n^2 + 9 is a 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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A000129, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114695, A114696, A114697.

I:=[1,6,13,36]; [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

Table[3*Fibonacci[n+1, 2] -1-(-1)^n, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)

Vec((-1-4*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^30)) \\ Colin Barker, May 26 2016

[3*lucas_number1(n+1,2,-1) -(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021

G.f.: (1 +4*x -x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
From Colin Barker, May 26 2016: (Start)
a(n) = (-1 - (-1)^n) + 3*((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) = 3*A000129(n+1) - (1 + (-1)^n). - G. C. Greubel, May 24 2021

cp's OEIS Frontend

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

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

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A114689 Expansion of (1 +4*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

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

A114689 Expansion of (1 +4x -x^2)/((1-x^2)(1-2*x-x^2)); a Pellian-related sequence.