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.

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A129920 Expansion of -1/(1 - x + 3*x^2 - 2*x^3 + x^4 - 2*x^5 + x^6).

Original entry on oeis.org

-1, -1, 2, 3, -4, -10, 5, 29, 2, -76, -45, 178, 212, -361, -750, 565, 2282, -306, -6206, -2428, 15176, 14353, -32719, -55104, 57933, 176234, -61524, -499047, -97429, 1271400, 921652, -2887641, -3948938, 5590078, 13380187, -7828378, -39536779, 108416, 104810904
Offset: 0

Views

Author

Roger L. Bagula, Jun 05 2007

Keywords

Links

Crossrefs

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712.
Cf. A125629, A129704, A129903.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) )); // G. C. Greubel, Sep 28 2024
  • Mathematica
    CoefficientList[Series[-1/(1-x +3*x^2 -2*x^3 +x^4 -2*x^5 +x^6), {x,0,50}], x]
  • SageMath
    def A129920_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) ).list()
A129920_list(50) # G. C. Greubel, Sep 28 2024

Formula

a(n) = a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A100049 A Chebyshev transform of the Padovan numbers.

Original entry on oeis.org

1, 0, -1, 1, -1, -3, 3, 3, -6, 2, 10, -13, -9, 29, -9, -43, 55, 32, -126, 48, 183, -243, -121, 541, -241, -765, 1082, 450, -2326, 1171, 3179, -4803, -1617, 9993, -5601, -13168, 21250, 5552, -42849, 26489, 54351, -93763, -17765, 183347, -124086, -223422, 412698, 49827, -782881, 576541, 914279
Offset: 0

Views

Author

Paul Barry, Oct 31 2004

Keywords

Comments

A Chebyshev transform of the Padovan numbers A000931(n+3): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

Links

Crossrefs

Cf. A099443, A011655, A100047, A100048.

Programs

  • Mathematica
    LinearRecurrence[{0, -2, 1, -2, 0, -1}, {1, 0, -1, 1, -1, -3, 3}, 50] (* G. C. Greubel, Aug 08 2017 *)
  • PARI
    x='x+O('x^50); Vec((1-x^2)*(1+x^2)^2/(1+2*x^2-x^3+2*x^4+x^6)) \\ G. C. Greubel, Aug 08 2017

Formula

G.f.: (1-x^2)*(1+x^2)^2/(1+2*x^2-x^3+2*x^4+x^6).
a(n) = -2*a(n-2) +a(n-3) -2*a(n-4) -a(n-6).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A000931(n-2*k+3)/(n-k).

A125629 Expansion of -1/(1 - x + x^2 - x^3 + x^4 + x^6).

Original entry on oeis.org

1, -1, 0, 0, 0, 1, 2, 2, 1, 0, -1, -3, -5, -5, -3, 0, 4, 9, 13, 13, 8, -1, -13, -26, -35, -34, -20, 6, 40, 74, 95, 89, 48, -26, -120, -209, -258, -232, -111, 98, 355, 587, 699, 601, 245, -342, -1040, -1641, -1887, -1545, -504, 1137, 3023, 4568, 5073, 3936, 912
Offset: 0

Views

Author

Roger L. Bagula, Jun 07 2007

Keywords

Links

Crossrefs

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712, A129920, A129704, A129903.

Programs

  • Mathematica
    CoefficientList[Series[-1/(1 - x + x^2 - x^3 + x^4 + x^6), {x, 0, 50}], x]

Formula

G.f.: 1/(x^(17/2)*f(x)), where f(x) = -1/x^(5/2) - 1/x^(9/2) + 1/x^(11/2) + -1/x^(13/2) + 1/x^(15/2) - 1/x^(17/2) is the Jones polynomial for the link with Dowker-Thistlethwaite notation L6a3.
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A332828 Expansion of (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8) in powers of x.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1
Offset: 0

Views

Author

Michael Somos, Feb 25 2020

Keywords

Comments

This is a (-1,1) generalized Somos-4 sequence.
For the elliptic curve y^2 + y = x^3 - x^2, the multiples of the point (0, 0) are (a(n-1)*a(n+1)/a(n)^2, -a(n-1)^2*a(n+2)/a(n)^3).

Examples

			G.f. = x + x^2 + x^3 + x^4 + x^6 - x^7 + x^8 - x^9 - x^11 - x^12 + ...

Links

Crossrefs

Cf. A006720, A099443.

Programs

  • Mathematica
    a[ n_] := {1, 1, 1, 1, 0, 1, -1, 1, -1, 0}[[Mod[n, 10, 1]]];
  • PARI
    {a(n) = (-1)^(n\10) * [0, 1, 1, 1, 1, 0, 1, -1, 1, -1][n%10 + 1]};
  • PARI
    {a(n) = my(E=ellinit([0, -1, 1, 0, 0]), z=ellpointtoz(E, [0, 0])); (-1)^(n\2) * round(ellsigma(E, n*z) / ellsigma(E, z)^n^2)};

Formula

G.f.: (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8).
a(n) = -a(n+10) = a(5-n) for all n in Z.
a(n) * a(n+4) = -a(n+1) * a(n+3) + a(n+2)^2 for all n in Z.
a(n) * a(n+5) = -a(n+1) * a(n+4) + a(n+2)*a(n+3) for all n in Z.
a(2*n) = A099443(n-1), a(2*n+1) = A099443(n+2) for all n in Z.

A128063 Hankel transform of A115962.

Original entry on oeis.org

1, 2, 0, -8, -16, -32, -64, 0, 256, 512, 1024, 2048, 0, -8192, -16384, -32768, -65536, 0, 262144, 524288, 1048576, 2097152, 0, -8388608, -16777216, -33554432, -67108864, 0, 268435456, 536870912, 1073741824, 2147483648, 0, -8589934592, -17179869184
Offset: 0

Views

Author

Paul Barry, Feb 13 2007

Keywords

Crossrefs

Cf. A099443, A115962.

Formula

a(n) = 2^n*((1/2 - 3*sqrt(5)/10)*cos(3*Pi*n/5) + sqrt(1/10 - sqrt(5)/50)*sin(3*Pi*n/5) + (3*sqrt(5)/10 + 1/2)*cos(Pi*n/5) - sqrt(sqrt(5)/50 + 1/10)*sin(Pi*n/5));
a(n) = 2^n*Sum_{k=0..floor((n+2)/2)} binomial(n-k+2,k)*(-1)^k*Fibonacci(n-2k+3);
a(n) = 2^n*A099443(n+2).
Empirical g.f.: -(2*x-1)*(4*x^2 + 2*x + 1) / (16*x^4 - 8*x^3 + 4*x^2 - 2*x + 1). - Colin Barker, Jun 28 2013
Previous Showing 11-15 of 15 results.

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