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.

A090301 a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.

Original entry on oeis.org

2, 15, 227, 3420, 51527, 776325, 11696402, 176222355, 2655031727, 40001698260, 602680505627, 9080209282665, 136805819745602, 2061167505466695, 31054318401746027, 467875943531657100, 7049193471376602527
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) = 0.066372... = 2/(15+sqrt(229)) = (sqrt(229)-15)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 15.066372... = (15+sqrt(229))/2 = 2/(sqrt(229)-15).
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) = 15*a(3) + a(2) = 15*3420 + 227 = ((15+sqrt(229))/2)^4 + ((15-sqrt(229))/2)^4 = 51526.9999805 + 0.0000194 = 51527.

Links

Crossrefs

Cf. A058087, A071416.
Lucas polynomials: A114525.
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), this sequence (m=15), A090305 (m=16), A090306 (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), A087281 (m=29), A087287 (m=76), A089772 (m=199).

Programs

  • GAP
    m:=15;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 31 2019
  • Magma
    m:=15; 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 31 2019
  • Maple
    seq(simplify(2*(-I)^n*ChebyshevT(n, 15*I/2)), n = 0..20); # G. C. Greubel, Dec 31 2019
  • Mathematica
    LucasL[Range[20]-1, 15] (* G. C. Greubel, Dec 31 2019 *)
  • PARI
    vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 15*I/2) ) \\ G. C. Greubel, Dec 31 2019
  • Sage
    [2*(-I)^n*chebyshev_T(n, 15*I/2) for n in (0..20)] # G. C. Greubel, Dec 31 2019

Formula

a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.
a(n) = ((15+sqrt(229))/2)^n + ((15-sqrt(229))/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.: (2-15*x)/(1-15*x-x^2). - Philippe Deléham, Nov 02 2008
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
Lim_{k-> infinity} a(n+k)/a(k) = (A090301(n) + A154597(n)*sqrt(229))/2.
Lim_{n-> infinity} A090301(n)/ A154597(n) = sqrt(229).
a(2n+1) = 15*A098246(n).
a(3n+1) = A041426(5n), a(3n+2) = A041426(5n+3), a(3n+3) = 2*A041426(5n+4).
(End)
a(n) = Lucas(n, 15) = 2*(-i)^n * ChebyshevT(n, 15*i/2). - G. C. Greubel, Dec 31 2019
E.g.f.: 2*exp(15*x/2)*cosh(sqrt(229)*x/2). - Stefano Spezia, Jan 01 2020

Extensions

More terms from Ray Chandler, Feb 14 2004

A098245 Chebyshev polynomials S(n,227).

Original entry on oeis.org

1, 227, 51528, 11696629, 2655083255, 602692202256, 136808474828857, 31054921093948283, 7049330279851431384, 1600166918605180975885, 363230841193096230094511, 82451800783914239050478112
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

Used for all positive integer solutions of Pell equation x^2 - 229*y^2 = -4. See A098246 with A098247.

Links

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - 227*x + x^2), {x, 0, 15}], x] (* Wesley Ivan Hurt, Feb 09 2017 *)
LinearRecurrence[{227,-1},{1,227},20] (* Harvey P. Dale, Jan 15 2019 *)

Formula

a(n) = S(n, 227) = U(n, 227/2) = S(2*n+1, sqrt(229))/sqrt(229) 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) = 227*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=227; a(-1):=0.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (227+15*sqrt(229))/2 and am := (227-15*sqrt(229))/2 = 1/ap.
G.f.: 1/(1-227*x+x^2).

A098247 First differences of Chebyshev polynomials S(n,227)=A098245(n) with Diophantine property.

Original entry on oeis.org

1, 226, 51301, 11645101, 2643386626, 600037119001, 136205782626601, 30918112619119426, 7018275358757483101, 1593117588325329544501, 361630674274491049118626, 82088569942721142820383601
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

(15*b(n))^2 - 229*a(n)^2 = -4 with b(n)=A098246(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 229*y^2 = -4 are (15=15*1,1), (3420=15*228,226), (776325=15*51755,51301), (176222355=15*11748157,11645101), ...

Links

Programs

  • GAP
    a:=[1,226];; for n in [3..20] do a[n]:=227*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
  • Magma
    I:=[1,226]; [n le 2 select I[n] else 227*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
  • Mathematica
    LinearRecurrence[{227,-1}, {1,226}, 20] (* G. C. Greubel, Aug 01 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1-x)/(1-227*x+x^2)) \\ G. C. Greubel, Aug 01 2019
  • Sage
    ((1-x)/(1-227*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = S(n, 227) - S(n-1, 227) = T(2*n+1, sqrt(229)/2)/(sqrt(229)/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 second kind, A053120.
a(n) = ((-1)^n)*S(2*n, 15*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-227*x+x^2).
a(n) = 227*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=226. - Philippe Deléham, Nov 18 2008
Showing 1-3 of 3 results.

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