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 1-4 of 4 results.

A088138 Generalized Gaussian Fibonacci integers.

Original entry on oeis.org

0, 1, 2, 0, -8, -16, 0, 64, 128, 0, -512, -1024, 0, 4096, 8192, 0, -32768, -65536, 0, 262144, 524288, 0, -2097152, -4194304, 0, 16777216, 33554432, 0, -134217728, -268435456, 0, 1073741824, 2147483648, 0, -8589934592, -17179869184, 0, 68719476736, 137438953472
Offset: 0

Views

Author

Paul Barry, Sep 20 2003

Keywords

Comments

The sequence 0,1,-2,0,8,-16,... has g.f. x/(1+2*x-4*x^2), a(n) = 2^n*sin(2n*Pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)*x)/sqrt(3): 0,1,-3,0,9,...
a(n+1) is the Hankel transform of A100192. - Paul Barry, Jan 11 2007
a(n+1) is the trinomial transform of A010892: a(n+1) = Sum_{k=0..2n} trinomial(n,k)*A010892(k+1) where trinomial(n, k) = trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n+1) is the Hankel transform of A100067. - Paul Barry, Jun 16 2009
From Paul Curtz, Oct 04 2009: (Start)
1) a(n) = A131577(n)*A128834(n).
2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244.
3) Sequence is identical to every 2n-th difference divided by (-3)^n.
4) a(3n) + a(3n+1) + a(3n+2) = (-1)^n*3*A001018(n) for n >= 1.
5) For missing terms in a(n) see A013731 = 4*A001018. (End)
The coefficient of i of Q^n, where Q is the quaternion 1+i+j+k. Due to symmetry, also the coefficients of j and of k. - Stanislav Sykora, Jun 11 2012 [The coefficients of 1 are in A138230. - Wolfdieter Lang, Jan 28 2016]
With different signs, 0, 1, -2, 0, 8, -16, 0, 64, -128, 0, 512, -1024, ... is the Lucas U(-2,4) sequence. - R. J. Mathar, Jan 08 2013

Links

Crossrefs

Cf. A084102, A088137, A045873, A088139, A138230.

Programs

  • GAP
    a:=[0,1];; for n in [3..40] do a[n]:=2*a[n-1]-4*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018
  • Magma
    I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2018
  • Maple
    M:= <<1+I,1+I>|>:
T:= <<-I/2,0>|<0,I/2>>:
seq(LinearAlgebra:-Trace(T.M^n),n=0..100); # Robert Israel, Jan 28 2016
  • Mathematica
    Join[{a=0,b=1},Table[c=2*b-4*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{2, -4}, {0, 1}, 40] (* Vincenzo Librandi, Jan 29 2016 *)
Table[2^(n-2)*((-1)^Quotient[n-1,3]+(-1)^Quotient[n,3]), {n,0,40}] (*Federico Provvedi,Apr 24 2022*)
  • PARI
    /* lists powers of any quaternion */
QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);} /* Stanislav Sykora, Jun 11 2012 */
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x+4*x^2))) \\ G. C. Greubel, Oct 22 2018
  • PARI
    a(n) = 2^(n-1)*polchebyshev(n-1, 2, 1/2); \\ Michel Marcus, May 02 2022
  • Sage
    [lucas_number1(n,2,4) for n in range(0, 39)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1-2*x+4*x^2).
E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3).
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1.
a(n) = ((1+i*sqrt(3))^n - (1-i*sqrt(3))^n)/(2*i*sqrt(3)).
a(n) = Im( (1+i*sqrt(3))^n/sqrt(3) ).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*(-3)^k.
From Paul Curtz, Oct 04 2009: (Start)
a(n) = a(n-1) + a(n-2) + 2*a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). (End)
E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3) = G(0)*x^2 where G(k)= 1 + (3*k+2)/(2*x - 32*x^5/(16*x^4 - 3*(k+1)*(3*k+2)*(3*k+4)*(3*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2012
G.f.: x/(1-2*x+4*x^2) = 2*x^2*G(0) where G(k)= 1 + 1/(2*x - 32*x^5/(16*x^4 - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 27 2012
a(n) = -2^(n-1)*Product_{k=1..n}(1 + 2*cos(k*Pi/n)) for n >= 1. - Peter Luschny, Nov 28 2019
a(n) = 2^(n-1) * U(n-1, 1/2), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Apr 24 2022

A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*3^(n-2*k).

Original entry on oeis.org

1, 3, 11, 36, 123, 408, 1370, 4560, 15235, 50760, 169326, 564336, 1881582, 6271632, 20907156, 69689376, 232304355, 774343560, 2581169510, 8603882160, 28679699578, 95598937008, 318663476076, 1062211351776, 3540705857998, 11802351958608, 39341178395660, 131137257852000
Offset: 0

Views

Author

Paul Barry, Nov 02 2004

Keywords

Comments

An inverse Chebyshev transform of x/(1-3*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by Vaclav Kotesovec, Dec 06 2012
Generally (for r>1), a(n) ~ (r + 1/r)^n. - Vaclav Kotesovec, Dec 06 2012

Links

Crossrefs

Cf. A027306, A100067, A100069.

Programs

  • Magma
    m:=3; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022
  • Mathematica
    CoefficientList[Series[2*x/(Sqrt[1-4*x^2]*(3*Sqrt[1-4*x^2] + 2*x-3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)
  • PARI
    my(x='x+O('x^66)); Vec(2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3))) \\ Joerg Arndt, May 12 2013
  • SageMath
    m=3; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022

Formula

G.f.: 2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*3^(n-k).
a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1+(-1)^(n-k))*3^k/2.
D-finite with recurrence 9*n*a(n) +12*(-3*n+1)*a(n-1) +4*(-4*n-1)*a(n-2) +48*(3*n-4)*a(n-3) +80*(-n+3)*a(n-4)=0. - R. J. Mathar, Nov 22 2012
a(n) ~ 10^n/3^n. - Vaclav Kotesovec, Dec 06 2012

A100069 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*4^(n-2*k).

Original entry on oeis.org

1, 4, 18, 76, 326, 1384, 5892, 25036, 106438, 452344, 1922588, 8170936, 34726940, 147589264, 627256088, 2665837516, 11329815878, 48151714264, 204644809932, 869740430056, 3696396920116, 15709686864304, 66766169526008, 283756220309176, 1205963937666076, 5125346734404784
Offset: 0

Views

Author

Paul Barry, Nov 02 2004

Keywords

Comments

An inverse Chebyshev transform of x/(1-4*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by Vaclav Kotesovec, Dec 06 2012
Generally (for r>1), a(n) ~ (r + 1/r)^n. - Vaclav Kotesovec, Dec 06 2012

Links

Crossrefs

Cf. A027306, A100067, A100068.

Programs

  • Magma
    m:=4; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022
  • Mathematica
    CoefficientList[Series[x/(Sqrt[1-4*x^2]*(2*Sqrt[1-4*x^2]+x-2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)
  • PARI
    my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2))) \\ Joerg Arndt, May 12 2013
  • SageMath
    m=4; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022

Formula

G.f.: x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2)). - corrected by Vaclav Kotesovec, Dec 06 2012
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*4^(n-2*k).
a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1 + (-1)^(n-k))*4^k/2.
8*n*a(n) = 2*(19*n-4)*a(n-1) + (15*n+2)*a(n-2) - 8*(19*n-23)*a(n-3) + 68*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 22 2012
a(n) ~ 17^n/4^n. - Vaclav Kotesovec, Dec 06 2012

A100087 Expansion of x/(sqrt(1-4*x^2) + x - 1).

Original entry on oeis.org

1, 2, 4, 10, 24, 60, 148, 370, 920, 2300, 5736, 14340, 35808, 89520, 223668, 559170, 1397496, 3493740, 8732920, 21832300, 54575888, 136439720, 341082504, 852706260, 2131706864, 5329267160, 13322959888, 33307399720, 83267756400, 208169391000, 520420803060, 1301052007650
Offset: 0

Views

Author

Paul Barry, Nov 03 2004

Keywords

Comments

Inverse Chebyshev transform of (1-x^2)/((1-2*x)*(1+x^2)), the g.f. of A100088, under the mapping g(x) -> (1/sqrt(1-4*x^2))*g(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. Equivalently, its image under the Chebyshev map A(x) -> ((1-x^2)/(1+x^2))*A(x/(1+x^2)) is A100088.
Transform of 1/(1-2*x) under the mapping g(x) -> g(x*c(x^2)). - Paul Barry, Jan 17 2005

Links

Crossrefs

Cf. A000108, A100067, A100088.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( x/(Sqrt(1-4*x^2) +x-1) )); // G. C. Greubel, Jul 08 2022
  • Mathematica
    CoefficientList[Series[x/(Sqrt[1-4*x^2]+x-1), {x, 0, 50}], x] (* Vaclav Kotesovec, Dec 06 2012 *)
  • PARI
    my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)+x-1)) \\ Joerg Arndt, May 12 2013
  • SageMath
    @CachedFunction
def A100067(n): return sum( binomial(n,k)*2^(n-2*k) for k in (0..(n//2)) )
def A100087(n): return (3/5)*A100067(n) + (1/5)*((1+(-1)^n) -2*I*(1-(-1)^n))*I^n*(-1)^floor(n/2)*binomial(n-1, floor(n/2))
[A100087(n) for n in (0..60)] # G. C. Greubel, Jul 08 2022

Formula

a(n) = Sum_{k=0..floor(n/2)} C(n, k)*(3*2^(n-2*k) + 2*cos(Pi*(n-2*k)/2) + 4*sin(Pi*(n-2*k)/2))/5.
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*A100088(n-2*k).
a(n) = Sum_{k=0..n} k*C(n-1,(n-k)/2)*(1 + (-1)^(n-k))*2^k/(n+k). - Paul Barry, Jan 17 2005
D-finite with recurrence: 4*n*a(n) + 2*(2*n-7)*a(n-1) - (51*n-83)*a(n-2) - 8*(2*n-13)*a(n-3) + 140*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 22 2012
a(n) ~ 3*5^(n-1)/2^n. - Vaclav Kotesovec, Dec 06 2012
Showing 1-4 of 4 results.

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