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.

A099486 Expansion of x/((1 + x^2)*(1 - 4*x + x^2)).

Original entry on oeis.org

0, 1, 4, 14, 52, 195, 728, 2716, 10136, 37829, 141180, 526890, 1966380, 7338631, 27388144, 102213944, 381467632, 1423656585, 5313158708, 19828978246, 74002754276, 276182038859, 1030725401160, 3846719565780, 14356152861960
Offset: 0

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Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence 0,1,4,16,... which has g.f. x/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).

Crossrefs

Cf. A099487, A099488.

Formula

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

A099488 Expansion of (1-x)^2/((1+x^2)(1-4x+x^2)).

Original entry on oeis.org

1, 2, 7, 28, 105, 390, 1455, 5432, 20273, 75658, 282359, 1053780, 3932761, 14677262, 54776287, 204427888, 762935265, 2847313170, 10626317415, 39657956492, 148005508553, 552364077718, 2061450802319, 7693439131560, 28712305723921
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence A081294 which has with g.f. (1-2x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

Crossrefs

Cf. A099486, A099487.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)^2/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{4,-2,4,-1},{1,2,7,28},30] (* Harvey P. Dale, Jun 23 2015 *)

Formula

a(n)=4a(n-1)-2a(n-2)+4a(n-3); a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3))}; a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))A001353(n-k)}; a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*(4^(n-2k)+0^(n-2k))/2}.

A099489 Expansion of (1-x+x^2)/((1+x^2)(1-4x+x^2)).

Original entry on oeis.org

1, 3, 11, 42, 157, 585, 2183, 8148, 30409, 113487, 423539, 1580670, 5899141, 22015893, 82164431, 306641832, 1144402897, 4270969755, 15939476123, 59486934738, 222008262829, 828546116577, 3092176203479, 11540158697340
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence A002001 which has with g.f. (1-x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

Crossrefs

Cf. A099486, A099487, A099488.

Programs

  • Mathematica
    CoefficientList[Series[(1-x+x^2)/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *_)
LinearRecurrence[{4,-2,4,-1},{1,3,11,42},30] (* Harvey P. Dale, Dec 28 2019 *)

Formula

a(n) = 4*a(n-1)-2*a(n-2)+4*a(n-3)-a(n-4). - corrected by Matthew House, Oct 22 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(3*4^(n-2*k)+0^(n-2*k))/4.
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3)).
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*A001353(n-k+1).
a(n) = 3*A001353(n+1)/4 +A056594(n)/4. - R. J. Mathar, Sep 21 2012
Showing 1-3 of 3 results.

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