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 1-5 of 5 results.

A114689 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, 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

Views

Author

Creighton Dement, Feb 18 2006

Keywords

Comments

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

Links

Crossrefs

Cf. A000129, A005409, A100828, A111954, A113224.
Cf. A114647, A114688, A114695, A114696, A114697.

Programs

  • Magma
    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
  • Mathematica
    Table[3*Fibonacci[n+1, 2] -1-(-1)^n, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)
  • PARI
    Vec((-1-4*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^30)) \\ Colin Barker, May 26 2016
  • Sage
    [3*lucas_number1(n+1,2,-1) -(1+(-1)^n) 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)).
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

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)

A111955 a(n) = A078343(n) + (-1)^n.

Original entry on oeis.org

0, 1, 4, 7, 20, 45, 112, 267, 648, 1561, 3772, 9103, 21980, 53061, 128104, 309267, 746640, 1802545, 4351732, 10506007, 25363748, 61233501, 147830752, 356895003, 861620760, 2080136521, 5021893804, 12123924127, 29269742060, 70663408245
Offset: 0

Views

Author

Creighton Dement, Aug 25 2005

Keywords

Comments

This sequence is a companion sequence to A111954 (compare formula / program code). Three other companion sequences (i.e., they are generated by the same floretion given in the program code) are A105635, A097076 and A100828.
Floretion Algebra Multiplication Program, FAMP Code: 4kbasejseq[J*D] with J = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and D = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e. (an initial term 0 was added to the sequence)

Links

Crossrefs

Cf. A078343, A000129, A001333, A111954, A111956, A007070, A077995, A100828, A097076, A105635, A048655.

Programs

  • Mathematica
    LinearRecurrence[{1,3,1},{0,1,4},40] (* Harvey P. Dale, Mar 12 2015 *)

Formula

a(n) + a(n+1) = A048655(n).
a(n) = a(n-1) + 3*a(n-2) + a(n-3), n >= 3; a(n) = (-1/4*sqrt(2)+1)*(1-sqrt(2))^n + (1/4*sqrt(2)+1)*(1+sqrt(2))^n - (-1)^n;
G.f.: -x*(1+3*x) / ( (1+x)*(x^2+2*x-1) ). - R. J. Mathar, Oct 02 2012
E.g.f.: cosh(x) - exp(x)*cosh(sqrt(2)*x) - sinh(x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, May 26 2024

A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).

Original entry on oeis.org

1, -1, 1, 1, 1, 1, -1, 1, 3, 1, 1, 1, 5, 5, 1, -1, 1, 7, 13, 7, 1, 1, 1, 9, 25, 25, 9, 1, -1, 1, 11, 41, 63, 41, 11, 1, 1, 1, 13, 61, 129, 129, 61, 13, 1, -1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1
Offset: 0

Views

Author

Paul Barry, Sep 07 2005

Keywords

Comments

Equivalent to Delannoy triangle A008288 with prepended column 1,-1,1,-1,... Row sums are A111954. Diagonal sums are A112476. Inverse is A112477.

Examples

			Triangle starts:
   1;
  -1, 1;
   1, 1, 1;
  -1, 1, 3,  1;
   1, 1, 5,  5,  1;
  -1, 1, 7, 13,  7, 1;
   1, 1, 9, 25, 25, 9, 1;
  ...

Crossrefs

Cf. A008288, A111954, A112476, A112477.

Programs

  • Mathematica
    T[n_,k_]:=SeriesCoefficient[(x(1+x)/(1-x))^k/(1+x),{x,0,n}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, May 26 2024 *)

Formula

T(n,k) = Sum{j=0..n-k} C(k-1,j)*C(n-j-1,n-k-j).
Showing 1-5 of 5 results.

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

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