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.

A077423 Chebyshev sequence U(n,12)=S(n,24) with Diophantine property.

Original entry on oeis.org

1, 24, 575, 13776, 330049, 7907400, 189447551, 4538833824, 108742564225, 2605282707576, 62418042417599, 1495427735314800, 35827847605137601, 858372914787987624, 20565122107306565375, 492704557660569581376
Offset: 0

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

b(n)^2 - 143*a(n)^2 = 1 with the companion sequence b(n)=A077424(n+1).
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 24's along the main diagonal, and i's along the subdiagonal and the superdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,23}. - Milan Janjic, Jan 25 2015

Links

Crossrefs

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077912 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), this sequence (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
Cf. A323182.

Programs

  • GAP
    a:=[1,24];; for n in [3..20] do a[n]:=24*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 1/(1-24*x+x^2) )); // G. C. Greubel, Dec 22 2019
  • Maple
    seq( simplify(ChebyshevU(n, 12)), n=0..20); # G. C. Greubel, Dec 22 2019
  • Mathematica
    Table[GegenbauerC[n, 1, 12], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
ChebyshevU[Range[21] -1, 12] (* G. C. Greubel, Dec 22 2019 *)
  • PARI
    vector(21, n, polchebyshev(n-1, 2, 12) ) \\ G. C. Greubel, Dec 22 2019
  • Sage
    [lucas_number1(n,24,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 24*a(n-1) - a(n-2), a(-1) = 0, a(0) = 1.
a(n) = S(n, 24) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap= 12+sqrt(143) and am = 12-sqrt(143).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*24^(n-2*k).
a(n) = sqrt((A077424(n+1)^2 - 1)/143).
G.f.: 1/(1-24*x+x^2). - Philippe Deléham, Nov 18 2008
a(n) = Sum_{k=0..n} A101950(n,k)*23^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/11*(11 + sqrt(143)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/24*(11 + sqrt(143)). - Peter Bala, Dec 23 2012

A077963 Expansion of 1/(1+x^2+2*x^3).

Original entry on oeis.org

1, 0, -1, -2, 1, 4, 3, -6, -11, 0, 23, 22, -23, -68, -21, 114, 157, -72, -385, -242, 529, 1012, -45, -2070, -1979, 2160, 6119, 1798, -10439, -14036, 6843, 34914, 21229, -48600, -91057, 6142, 188257, 175972, -200541, -552486, -151403, 953568, 1256375, -650762, -3163511, -1861988, 4465035
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A077912.

Programs

  • GAP
    a:=[1,0,-1];; for n in [4..50] do a[n]:=-a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 23 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+x^2+2*x^3) )); // G. C. Greubel, Jun 23 2019
  • Mathematica
    CoefficientList[Series[1/(1+x^2+2*x^3), {x,0,50}], x] (* or *) LinearRecurrence[{0,-1,-2}, {1,0,-1}, 50] (* G. C. Greubel, Jun 23 2019 *)
  • PARI
    my(x='x+O('x^50)); Vec(1/(1+x^2+2*x^3)) \\ G. C. Greubel, Jun 23 2019
  • Sage
    (1/(1+x^2+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 23 2019

Formula

a(n) = (-1)^n * A077912(n). - G. C. Greubel, Jun 23 2019

A110291 Riordan array (1/(1-x), x*(1+2*x)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 9, 7, 1, 1, 3, 9, 19, 9, 1, 1, 3, 9, 27, 33, 11, 1, 1, 3, 9, 27, 65, 51, 13, 1, 1, 3, 9, 27, 81, 131, 73, 15, 1, 1, 3, 9, 27, 81, 211, 233, 99, 17, 1, 1, 3, 9, 27, 81, 243, 473, 379, 129, 19, 1, 1, 3, 9, 27, 81, 243, 665, 939, 577, 163, 21, 1
Offset: 0

Views

Author

Paul Barry, Jul 18 2005

Keywords

Comments

Inverse is A110292.

Examples

			Rows begin
  1;
  1, 1;
  1, 3, 1;
  1, 3, 5,  1;
  1, 3, 9,  7,  1;
  1, 3, 9, 19,  9,   1;
  1, 3, 9, 27, 33,  11,  1;
  1, 3, 9, 27, 65,  51, 13,  1;
  1, 3, 9, 27, 81, 131, 73, 15, 1;

Links

Crossrefs

Cf. A000244, A001045, A001047, A005408, A014335.
Cf. A066810, A077890, A077912, A110292, A180146.
Cf. A000975 (row sums), A052947 (diagonal sums).

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
F:= func< k | Coefficients(R!( x^k*(1+2*x)^k/(1-x) )) >;
A110291:= func< n,k | F(k)[n-k+1] >;
[A110291(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 05 2023
  • Mathematica
    F[k_]:= CoefficientList[Series[x^k*(1+2*x)^k/(1-x), {x,0,40}], x];
A110291[n_, k_]:= F[k][[n+1]];
Table[A110291[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2023 *)
  • SageMath
    def p(k,x): return x^k*(1+2*x)^k/(1-x)
def A110291(n,k): return ( p(k,x) ).series(x, 30).list()[n]
flatten([[A110291(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2023

Formula

T(n, k) = [x^n]( x^k*(1+2*x)^k/(1-x) ).
Sum_{k=0..n} T(n, k) = A000975(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A052947(n+1).
From G. C. Greubel, Jan 05 2023: (Start)
T(n, 0) = T(n, n) = 1.
T(n, n-1) = A005408(n-1).
T(2*n, n) = T(2*n+1, n) = A000244(n).
T(2*n, n+1) = A066810(n+1).
T(2*n, n-1) = A000244(n-1).
T(2*n+1, n+1) = A001047(n+1).
Sum_{k=0..n} (-1)^k * T(n, k) = A077912(n).
Sum_{k=0..n} 2^k * T(n, k) = A014335(n+2).
Sum_{k=0..n} 3^k * T(n, k) = A180146(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A077890(n). (End)

Extensions

a(30) and following corrected by Georg Fischer, Oct 11 2022
Showing 1-3 of 3 results.

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