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 1-3 of 3 results.

A110047 Expansion of (1+2*x-4*x^2)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).

Original entry on oeis.org

1, 6, 28, 144, 688, 3360, 16192, 78336, 378112, 1826304, 8817664, 42577920, 205582336, 992649216, 4792926208, 23142334464, 111741042688, 539533639680, 2605098729472, 12578530000896, 60734514921472, 293252181786624, 1415946786832384, 6836795882864640
Offset: 0

Views

Author

Creighton Dement, Jul 10 2005

Keywords

Comments

Note (see program code): ibaseseq[A*B] = A057087, basejseq[A*B] = A099582, tesseq[A*B] = A110046.

Links

Crossrefs

Cf. A110046, A110048, A110049.

Programs

  • Maple
    seriestolist(series((1+2*x-4*x^2)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -kbasekseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'
  • Mathematica
    CoefficientList[Series[(1 + 2 x - 4 x^2)/((2 x + 1)(2 x - 1)(4 x^2 + 4 x - 1)), {x, 0, 21}], x] (* or *)
LinearRecurrence[{4, 8, -16, -16}, {1, 6, 28, 144}, 22] (* Michael De Vlieger, Feb 17 2017 *)
  • PARI
    Vec((1+2*x-4*x^2) / ((2*x+1)*(2*x-1)*(4*x^2+4*x-1)) + O(x^30)) \\ Colin Barker, Feb 17 2017

Formula

a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4). - Matthew House, Feb 17 2017
a(n) = (-3*(2-2*sqrt(2))^n*(-2+sqrt(2)) + 2^n*(-2*(1+(-1)^n)+3*(1+sqrt(2))^n*(2+sqrt(2)))) / 8. - Colin Barker, Feb 17 2017

Extensions

Definition corrected by Matthew House, Feb 17 2017

A099581 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*3^(n-k-1).

Original entry on oeis.org

0, 0, 1, 3, 15, 54, 216, 810, 3105, 11745, 44631, 169128, 641520, 2431944, 9221121, 34959195, 132543135, 502506990, 1905156936, 7222991778, 27384465825, 103822372809, 393620574951, 1492328843280, 5657848431840, 21450531825360
Offset: 0

Views

Author

Paul Barry, Oct 23 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*r^(n-k-1) has g.f. x^2/((1-r*x^2)*(1-r*x-r*x^2)) and satisfies a(n) = r*a(n-1) + 2*r*a(n-2) - r^2*a(n-3) - r^2*a(n-4).

Links

Crossrefs

Cf. A030195, A099177, A099582.

Programs

  • Magma
    [n le 4 select Floor((n-1)^2/3) else 3*Self(n-1) +6*Self(n-2) -9*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 23 2022
  • Mathematica
    LinearRecurrence[{3,6,-9,-9},{0,0,1,3},40] (* Harvey P. Dale, Jun 07 2021 *)
  • SageMath
    @CachedFunction
def a(n):
    if (n<4): return floor(n^2/3)
    else: return 3*a(n-1) + 6*a(n-2) - 9*a(n-3) - 9*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 23 2022

Formula

G.f.: x^2/((1-3*x^2)*(1-3*x-3*x^2)).
a(n) = 3*a(n-1) + 6*a(n-2) - 9*a(n-3) - 9*a(n-4).
From G. C. Greubel, Jul 23 2022: (Start)
a(n) = (2*(-i*sqrt(3))^(n-1)*ChebyshevU(n-1, i*sqrt(3)/2) - (1-(-1)^n)*3^((n - 1)/2))/6.
E.g.f.: (4*exp(3*x/2)*sinh(sqrt(21)*x/2) - 2*sqrt(7)*sinh(sqrt(3)*x))/(6*sqrt(21)). (End)

A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.

Original entry on oeis.org

0, 1, 8, 53, 316, 1785, 9744, 51997, 273092, 1417889, 7299160, 37334661, 190028748, 963565513, 4871514656, 24572321645, 123720601684, 622038982257, 3123938806632, 15674669614549, 78593250398300, 393845861293721
Offset: 0

Views

Author

Paul Barry, Oct 25 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1) * u^(n-k-1) * (v/u)^(k-1) has g.f. x^2/((1-u*x) * (1-u*x-v*x^2)) and satisfies the recurrence a(n) = 2*u*a(n-1) - (u^2 - v)*a(n-2) - u*v*a(n-3).

Links

Crossrefs

Cf. A094705, A099582, A099621.

Programs

  • Magma
    [(5^(n+2) -6*4^(n+1) -(-1)^n)/30: n in [0..40]]; // G. C. Greubel, Jul 22 2022
  • Mathematica
    LinearRecurrence[{8,-11,-20},{0,1,8},30] (* Harvey P. Dale, Nov 05 2017 *)
  • SageMath
    [(5^(n+2) -6*4^(n+1) -(-1)^n)/30 for n in (0..40)] # G. C. Greubel, Jul 22 2022

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.
a(n) = 8*a(n-1) - 11*a(n-2) - 20*a(n-3).
G.f.: x^2/((1-4*x)*(1-4*x-5*x^2)) = x^2/((1+x)*(1-4*x)*(1-5*x)).
From G. C. Greubel, Jul 22 2022: (Start)
a(n) = (1/30)*(5^(n+2) - 6*4^(n+1) - (-1)^n).
E.g.f.: (1/30)*(25*exp(5*x) - 24*exp(4*x) - exp(-x)). (End)
Showing 1-3 of 3 results.

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