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 1-5 of 5 results.

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

A162562 a(n) = ((5+sqrt(3))*(1+sqrt(3))^n + (5-sqrt(3))*(1-sqrt(3))^n)/2.

Original entry on oeis.org

5, 8, 26, 68, 188, 512, 1400, 3824, 10448, 28544, 77984, 213056, 582080, 1590272, 4344704, 11869952, 32429312, 88598528, 242055680, 661308416, 1806728192, 4936073216, 13485602816, 36843352064, 100657909760, 275002523648
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

Keywords

Comments

Binomial transform of A162813. Inverse binomial transform of A162563.

Links

Crossrefs

Cf. A162813, A162563.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r)*(1+r)^n+(5-r)*(1-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 14 2009
  • Mathematica
    LinearRecurrence[{2,2},{5,8},30] (* Harvey P. Dale, Aug 17 2013 *)

Formula

a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0) = 5, a(1) = 8.
G.f.: (5-2*x)/(1-2*x-2*x^2).

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 14 2009

A162563 a(n) = ((5+sqrt(3))*(2+sqrt(3))^n + (5-sqrt(3))*(2-sqrt(3))^n)/2.

Original entry on oeis.org

5, 13, 47, 175, 653, 2437, 9095, 33943, 126677, 472765, 1764383, 6584767, 24574685, 91713973, 342281207, 1277410855, 4767362213, 17792037997, 66400789775, 247811121103, 924843694637, 3451563657445, 12881410935143, 48074080083127
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

Keywords

Comments

Binomial transform of A162562. Second binomial transform of A162813. Inverse binomial transform of A162814.

Links

Crossrefs

Cf. A162562, A162813, A162814.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r)*(2+r)^n+(5-r)*(2-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 14 2009
  • Mathematica
    LinearRecurrence[{4,-1},{5,13},30] (* Harvey P. Dale, Aug 25 2014 *)

Formula

a(n) = 4*a(n-1) - a(n-2) for n > 1; a(0) = 5, a(1) = 13.
G.f.: (5-7*x)/(1-4*x+x^2).
a(n) = 4*a(n-1) - a(n-2), with a(0)=5 and a(1)=13. - Paolo P. Lava, Jul 15 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 18 2009

A162852 a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1.

Original entry on oeis.org

3, -1, 9, -3, 27, -9, 81, -27, 243, -81, 729, -243, 2187, -729, 6561, -2187, 19683, -6561, 59049, -19683, 177147, -59049, 531441, -177147, 1594323, -531441, 4782969, -1594323, 14348907, -4782969, 43046721, -14348907, 129140163
Offset: 1

Views

Author

Klaus Brockhaus, Jul 14 2009

Keywords

Comments

Third binomial transform is A162560.
Equivalently, 3^n followed by -3^(n-1), n > 0. - Muniru A Asiru, Oct 25 2018

Links

Crossrefs

Cf. A162560, A162436, A162766, A162813.

Programs

  • GAP
    a:=[3,-1];; for n in [3..25] do a[n]:=3*a[n-2]; od; a; # Muniru A Asiru, Oct 25 2018
  • Magma
    [ n le 2 select 7-4*n else 3*Self(n-2): n in [1..34] ];
  • Maple
    seq(op([3^n,-3^(n-1)]),n=1..18); # Muniru A Asiru, Oct 25 2018
  • Mathematica
    Rest[CoefficientList[Series[x*(3-x)/(1-3*x^2), {x, 0, 40}], x]] (* or *) LinearRecurrence[{0,3}, {3,-1}, 40] (* G. C. Greubel, Oct 24 2018 *)
  • PARI
    x='x+O('x^40); Vec(x*(3-x)/(1-3*x^2)) \\ G. C. Greubel, Oct 24 2018

Formula

a(n) = ((4-5*(-1)^n)*3^(1/4*(2*n-1+(-1)^n)))/3.
G.f.: x*(3-x)/(1-3*x^2). [corrected by Klaus Brockhaus, Sep 18 2009]
E.g.f.: (1 - cosh(sqrt(3)*x) + 3*sqrt(3)*sinh(sqrt(3)*x))/3. - G. C. Greubel, Oct 24 2018

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 1-5 of 5 results.

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