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 1-4 of 4 results.

A162766 a(n) = 3*a(n-2) for n > 2; a(1) = 4, a(2) = 3.

Original entry on oeis.org

4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163
Offset: 1

Views

Author

Klaus Brockhaus, Jul 13 2009

Keywords

Comments

Binomial transform is A162559. Second binomial transform is A077236.

Links

Crossrefs

Cf. A074324, A166552, A162559, A077236, A162436.

Programs

  • Magma
    [ n le 2 select 5-n else 3*Self(n-2): n in [1..34] ];
  • PARI
    a(n)=3^(n\2)*4^(n%2) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = (5-3*(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(4+3*x)/(1-3*x^2).
a(n) = A074324(n+1) = A166552(n+1) = 3^floor(n/2)*4^(n%2), where n%2 = 0 for n even, 1 for n odd. - M. F. Hasler, Dec 03 2014

Extensions

G.f. corrected by Klaus Brockhaus, Sep 18 2009

A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808
Offset: 1

Views

Author

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

Keywords

Comments

Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.
First differences are in A161728.
Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....

Links

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ];  // Klaus Brockhaus, Dec 31 2008
  • Magma
    I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016
  • Mathematica
    Join[{a=1,b=8},Table[c=8*b-13*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-13},{1,8},40] (* Harvey P. Dale, Aug 16 2012 *)
  • PARI
    a(n)=([0,1; -13,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 04 2016
  • Sage
    [lucas_number1(n,8,13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = 8*a(n-1) - 13*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(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016
a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016
a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016

Extensions

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

A166552 a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4.

Original entry on oeis.org

1, 4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163
Offset: 1

Views

Author

Klaus Brockhaus, Oct 16 2009

Keywords

Comments

Interleaving of A000244 (powers of 3) and 4*A000244.
a(n) = A074324(n); A074324 has the additional term a(0)=1.
First differences are in A162852.
Second binomial transform is A054491. Fourth binomial transform is A153594.

Links

Crossrefs

Equals A162766 preceded by 1.
Cf. A000244 (powers of 3), A074324, A162852, A054491, A153594.

Programs

  • Magma
    [ n le 2 select 3*n-2 else 3*Self(n-2): n in [1..35] ];
  • Mathematica
    LinearRecurrence[{0, 3}, {1, 4}, 50] (* G. C. Greubel, May 17 2016 *)
  • PARI
    a(n)=3^(n\2)*(4/3)^!bittest(n,0) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = (7+(-1)^n)*3^(1/4*(2*n-5+(-1)^n))/2.
G.f.: x*(1+4*x)/(1-3*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 3^floor((n-1)/2)*4^(1-n%2). - M. F. Hasler, Dec 03 2014
E.g.f.: (sqrt(3)*sinh(sqrt(3)*x) + 4*cosh(sqrt(3)*x) - 4)/3. - Ilya Gutkovskiy, May 17 2016

A162466 a(n) = 12*a(n-2) for n > 2; a(1) = 1, a(2) = 8.

Original entry on oeis.org

1, 8, 12, 96, 144, 1152, 1728, 13824, 20736, 165888, 248832, 1990656, 2985984, 23887872, 35831808, 286654464, 429981696, 3439853568, 5159780352, 41278242816, 61917364224, 495338913792, 743008370688, 5944066965504
Offset: 1

Views

Author

Klaus Brockhaus, Jul 04 2009

Keywords

Comments

Eighth binomial transform is A161729.

Links

Crossrefs

Cf. A161729, A161728, A162436, A162272, A074324.

Programs

  • Mathematica
    LinearRecurrence[{0,12},{1,8},30] (* Harvey P. Dale, Sep 17 2020 *)
  • PARI
    {m=24; v=concat([1, 8], vector(m-2)); for(n=3, m, v[n]=12*v[n-2]); v}
  • PARI
    Vec(x*(1+8*x)/(1-12*x^2)+O(x^29)) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = (5-(-1)^n)*2^(1/2 *(2*n-3+(-1)^n))*3^(1/4*(2*n-5+(-1)^n)).
G.f.: x*(1+8*x)/(1-12*x^2).
a(n) = 2^(n-1)*A074324(n). - M. F. Hasler, Dec 03 2014

Extensions

G.f. and comment corrected, formula added by Klaus Brockhaus, Sep 18 2009
Showing 1-4 of 4 results.

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