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

A058005 a(n) = gcd(2*n, binomial(2*n, n)).

Original entry on oeis.org

2, 2, 2, 2, 2, 12, 2, 2, 2, 4, 2, 4, 2, 4, 30, 2, 2, 12, 2, 20, 6, 4, 2, 12, 2, 4, 2, 56, 2, 4, 2, 2, 6, 4, 14, 4, 2, 4, 2, 20, 2, 84, 2, 8, 90, 4, 2, 12, 2, 4, 6, 8, 2, 12, 2, 8, 6, 4, 2, 24, 2, 4, 6, 2, 10, 132, 2, 4, 6, 20, 2, 36, 2, 4, 30, 8, 154, 12, 2, 20, 2, 4, 2, 56, 10, 4, 6, 88, 2, 20
Offset: 1

Views

Author

Labos Elemer, Nov 13 2000

Keywords

Comments

a(n) = 2 for values like 1,2,3,4,5,7, ...; a(n) = 2n for values like 6,15,...,190.

Links

Crossrefs

Cf. A001405, A071416.

Programs

  • Mathematica
    a[n_] := GCD[2*n, Binomial[2*n, n]]; Array[a, 100] (* Amiram Eldar, Mar 07 2025 *)
  • PARI
    a(n) = gcd(2*n, binomial(2*n, n)); \\ Amiram Eldar, Mar 07 2025

Extensions

Name corrected by Hugo Pfoertner, Mar 22 2020

A333461 a(n) = gcd(2*n, binomial(2*n,n))/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 2, 15, 1, 1, 6, 1, 10, 3, 2, 1, 6, 1, 2, 1, 28, 1, 2, 1, 1, 3, 2, 7, 2, 1, 2, 1, 10, 1, 42, 1, 4, 45, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 1, 5, 66, 1, 2, 3, 10, 1, 18, 1, 2, 15, 4, 77, 6, 1, 10, 1, 2
Offset: 1

Views

Author

Hugo Pfoertner, Mar 22 2020

Keywords

Links

Crossrefs

Cf. A058005, A071416.

Programs

  • PARI
    for(n=1,82,print1(gcd(2*n,binomial(2*n,n))/2,", "))

Formula

a(n) = A058005(n)/2.
Showing 1-3 of 3 results.

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