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.

A090306 a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

Original entry on oeis.org

2, 17, 291, 4964, 84679, 1444507, 24641298, 420346573, 7170533039, 122319408236, 2086600473051, 35594527450103, 607193567124802, 10357885168571737, 176691241432844331, 3014108989526925364, 51416544063390575519
Offset: 0

Views

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Keywords

Comments

Lim_{n-> infinity} a(n)/a(n+1) = 0.058621... = 2/(17+sqrt(293)) = (sqrt(293)-17)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 17.058621... = (17+sqrt(293))/2 = 2/(sqrt(293)-17).
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

Examples

			a(4) = 17*a(3) + a(2) = 17*4964 + 291=((17+sqrt(293))/2)^4 + ((17-sqrt(293))/2)^4 = 84678.999988190 + 0.000011809 = 84679.

Links

Crossrefs

Cf. A005074.
Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), this sequence (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

Programs

  • GAP
    m:=17;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 30 2019
  • Magma
    m:=17; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 30 2019
  • Maple
    seq(simplify(2*(-I)^n*ChebyshevT(n, 17*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019
  • Mathematica
    LinearRecurrence[{17,1},{2,17},30] (* Harvey P. Dale, Jan 24 2018 *)
LucasL[Range[20]-1, 17] (* G. C. Greubel, Dec 30 2019 *)
  • PARI
    vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 17*I/2) ) \\ G. C. Greubel, Dec 30 2019
  • Sage
    [2*(-I)^n*chebyshev_T(n, 17*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

Formula

a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.
a(n) = ((17+sqrt(293))/2)^n + ((17-sqrt(293))/2)^n.
(a(n))^2 = a(2n) - 2 if n=1, 3, 5, ...
(a(n))^2 = a(2n) + 2 if n=2, 4, 6, ...
G.f.: (2-17*x)/(1-17*x-x^2). - Philippe Deléham, Nov 02 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 17*A098249(n).
a(3n+1) = A041550(5n), a(3n+2) = A041550(5n+3), a(3n+3) = 2*A041550(5n+4).
Lim_{k-> infinity} a(n+k)/a(k) = (A090306(n) + A178765(n)*sqrt(293))/2.
Lim_{n-> infinity} A090306(n)/A178765(n) = sqrt(293). (End)
a(n) = Lucas(n, 17) = 2*(-i)^n * ChebyshevT(n, 17*i/2). - G. C. Greubel, Dec 30 2019
E.g.f.: 2*exp(17*x/2)*cosh(sqrt(293)*x/2). - Stefano Spezia, Dec 31 2019

Extensions

More terms from Ray Chandler, Feb 14 2004

A098250 First differences of Chebyshev polynomials S(n,291)=A098248(n) with Diophantine property.

Original entry on oeis.org

1, 290, 84389, 24556909, 7145976130, 2079454496921, 605114112627881, 176086127320216450, 51240457936070359069, 14910797173269154272629, 4338990736963387822975970, 1262631393659172587331734641, 367421396564082259525711804561, 106918363768754278349394803392610
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

(17*b(n))^2 - 293*a(n)^2 = -4 with b(n)=A098249(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 293*y^2 = -4 are (17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389), (420346573=17*24726269,24556909), ...

Links

Programs

  • GAP
    a:=[1,290];; for n in [3..20] do a[n]:=291*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
  • Magma
    I:=[1,290]; [n le 2 select I[n] else 291*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
  • Mathematica
    LinearRecurrence[{291,-1}, {1,290}, 20] (* G. C. Greubel, Aug 01 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1-x)/(1-291*x+x^2)) \\ G. C. Greubel, Aug 01 2019
  • Sage
    ((1-x)/(1-291*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = ((-1)^n)*S(2*n, 17*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-291*x+x^2).
a(n) = S(n, 291) - S(n-1, 291) = T(2*n+1, sqrt(293)/2)/(sqrt(293)/2), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n) = 291*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=290. - Philippe Deléham, Nov 18 2008

A098248 Chebyshev polynomials S(n,291).

Original entry on oeis.org

1, 291, 84680, 24641589, 7170617719, 2086625114640, 607200737742521, 176693328057958971, 51417151264128318040, 14962214324533282590669, 4353952951287921105566639, 1266985346610460508437301280
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

Used for all positive integer solutions of Pell equation x^2 - 293*y^2 = -4. See A098249 with A098250.

Links

Programs

  • Mathematica
    LinearRecurrence[{291,-1},{1,291},20] (* Harvey P. Dale, Dec 27 2015 *)

Formula

a(n)= S(n, 291)=U(n, 291/2)= S(2*n+1, sqrt(293))/sqrt(293) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n)=291*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=291; a(-1):=0.
a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (291+17*sqrt(293))/2 and am := (291-17*sqrt(293))/2 = 1/ap.
G.f.: 1/(1-291*x+x^2).
Showing 1-3 of 3 results.

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