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.

A055845 a(n) = 4*a(n-1) - a(n-2) with a(0)=1, a(1)=8.

Original entry on oeis.org

1, 8, 31, 116, 433, 1616, 6031, 22508, 84001, 313496, 1169983, 4366436, 16295761, 60816608, 226970671, 847066076, 3161293633, 11798108456, 44031140191, 164326452308, 613274669041, 2288772223856
Offset: 0

Views

Author

Barry E. Williams, May 31 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A054485.

Programs

  • GAP
    a:=[1,8];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 20 2020
  • Magma
    I:=[1,8]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 20 2020
  • Maple
    seq( simplify(ChebyshevU(n,2) + 4*ChebyshevU(n-1,2)), n=0..30); # G. C. Greubel, Jan 20 2020
  • Mathematica
    LinearRecurrence[{4,-1}, {1,8}, 30]  (* Sture Sjöstedt, Nov 30 2011 *)
Table[ChebyshevU[n, 2] + 4*ChebyshevU[n-1, 2], {n,0,30}] (* G. C. Greubel, Jan 20 2020 *)
  • PARI
    a(n) = polchebyshev(n,2,2) + 4*polchebyshev(n-1,2,2); \\ G. C. Greubel, Jan 20 2020
  • Sage
    [chebyshev_U(n,2) +4*chebyshev_U(n-1,2) for n in (0..30)] # G. C. Greubel, Jan 20 2020

Formula

a(n) = (8*((2+sqrt(3))^n - (2-sqrt(3))^n) - ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/(2*sqrt(3)).
G.f.: (1+4*x)/(1-4*x+x^2).
a(n)^2 = 3*A144721(n)^2 - 11. - Sture Sjöstedt, Nov 30 2011
From G. C. Greubel, Jan 20 2020: (Start)
a(n) = ChebyshevU(n,2) + 4*ChebyshevU(n-1,2).
E.g.f.: exp(2*x)*( cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) ). (End)

A153596 a(n) = ((5 + sqrt(3))^n - (5 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 10, 78, 560, 3884, 26520, 179752, 1214080, 8186256, 55152800, 371430368, 2500942080, 16837952704, 113358801280, 763153053312, 5137636904960, 34587001876736, 232842006858240, 1567506027294208, 10552536122060800, 71040228620135424
Offset: 1

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Keywords

Comments

Third binomial transform of A054485. Fifth binomial transform of A162813 preceded by 1.
Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(3) = 6.73205080756887729....

Links

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A054485, A162813.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
  • Magma
    I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-22*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016
  • Mathematica
    Table[Simplify[((5+Sqrt[3])^n -(5-Sqrt[3])^n)/(2*Sqrt[3])], {n,1,25}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011, modified by G. C. Greubel, Jun 01 2019 *)
LinearRecurrence[{10,-22},{1,10},25] (* G. C. Greubel, Aug 22 2016 *)
  • PARI
    my(x='x+O('x^25)); Vec(x/(1-10*x+22*x^2)) \\ G. C. Greubel, Jun 01 2019
  • Sage
    [lucas_number1(n,10,22) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

Formula

G.f.: x/(1 - 10*x + 22*x^2). - Klaus Brockhaus, Dec 31 2008 [corrected Oct 11 2009]
a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: sinh(sqrt(3)*x)*exp(5*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

Extensions

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008
Edited by Klaus Brockhaus, Oct 11 2009

A166465 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

Original entry on oeis.org

1, 5, 3, 15, 9, 45, 27, 135, 81, 405, 243, 1215, 729, 3645, 2187, 10935, 6561, 32805, 19683, 98415, 59049, 295245, 177147, 885735, 531441, 2657205, 1594323, 7971615, 4782969, 23914845, 14348907, 71744535, 43046721, 215233605, 129140163
Offset: 1

Views

Author

Klaus Brockhaus, Oct 14 2009

Keywords

Comments

Interleaving of A000244 and A005030.
Second binomial transform is A054485.
Fifth binomial transform is A153596.

Links

Crossrefs

Cf. A000244 (powers of 3), A005030 (5*3^n), A054485, A153596, A162813.

Programs

  • Magma
    [ n le 2 select 4*n-3 else 3*Self(n-2): n in [1..35] ];
  • Mathematica
    LinearRecurrence[{0,3}, {1,5}, 41] (* G. C. Greubel, Jul 27 2024 *)
  • SageMath
    [3^(n/2)*(5*((n+1)%2) +sqrt(3)*(n%2))/3 for n in range(1,41)] # G. C. Greubel, Jul 27 2024

Formula

a(n) = (4 + (-1)^n) * 3^((2*n - 5 + (-1)^n)/4).
G.f.: x*(1+5*x)/(1-3*x^2).
a(n) = A162813(n-1), for n >= 2.
From G. C. Greubel, Jul 27 2024: (Start)
a(n) = (1/6)*3^(n/2)*( 5*(1+(-1)^n) + sqrt(3)*(1-(-1)^n) ).
E.g.f.: (1/3)*(sqrt(3)*sinh(sqrt(3)*x) + 10*(sinh(sqrt(3)*x/2))^2). (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.