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.

A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467
Offset: 1

Views

Author

Klaus Brockhaus, Jul 03 2009, Jul 05 2009

Keywords

Comments

Interleaving of A000244 and 3*A000244.
Unsigned version of A128019.
Partial sums are in A164123.
Apparently a(n) = A056449(n-1) for n > 1. a(n) = A108411(n) for n >= 1.
Binomial transform is A026150 without initial 1, second binomial transform is A001834, third binomial transform is A030192, fourth binomial transform is A161728, fifth binomial transform is A162272.

Links

Crossrefs

Cf. A000244 (powers of 3), A128019 (expansion of (1-3x)/(1+3x^2)), A164123, A026150, A001834, A030192, A161728, A162272.
Essentially the same as A056449 (3^floor((n+1)/2)) and A108411 (powers of 3 repeated).

Programs

  • Magma
    [ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..35] ];
  • Mathematica
    CoefficientList[Series[(-3*x - 1)/(3*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]] (* or *) With[{c= 3^Range[20]},Join[{1},Riffle[c,c]]](* Harvey P. Dale, Feb 17 2012 *)
  • PARI
    a(n)=3^(n>>1) \\ Charles R Greathouse IV, Jul 15 2011

Formula

a(n) = 3^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 3*x)/(1 - 3*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
E.g.f.: cosh(sqrt(3)*x) - 1 + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

Extensions

G.f. corrected, formula simplified, comments added 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

A162272 a(n) = ((1+sqrt(3))*(5+sqrt(3))^n + (1-sqrt(3))*(5-sqrt(3))^n)/2.

Original entry on oeis.org

1, 8, 58, 404, 2764, 18752, 126712, 854576, 5758096, 38780288, 261124768, 1758081344, 11836068544, 79682895872, 536435450752, 3611330798336, 24311728066816, 163668003104768, 1101822013577728, 7417524067472384, 49935156376013824, 336166034275745792, 2263086902485153792
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

Comments

Fifth binomial transform of A162436, binomial transform of A161728.

Links

Crossrefs

Cf. A162436, A161728.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((1+r)*(5+r)^n+(1-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
  • Maple
    seq(expand(((1+sqrt(3))*(5+sqrt(3))^n+(1-sqrt(3))*(5-sqrt(3))^n)*1/2), n = 0 .. 20); # Emeric Deutsch, Jul 05 2009
  • Mathematica
    LinearRecurrence[{10, -22}, {1, 8}, 40] (* Vincenzo Librandi, Feb 03 2018 *)

Formula

From Emeric Deutsch, Jul 05 2009: (Start)
G.f.: (1 - 2*x)/(1 - 10*x + 22*x^2).
a(n) = 10*a(n-1) - 22*a(n-2) for n >= 2; a(0)=1, a(1)=8. (End)
E.g.f.: exp(5*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 05 2009
Extended by Emeric Deutsch, Jul 05 2009

A161729 a(n) = ((4+sqrt(3))*(8+2*sqrt(3))^n-(4-sqrt(3))*(8-2*sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 16, 204, 2432, 28304, 326400, 3750592, 43036672, 493555968, 5658988544, 64878906368, 743795097600, 8527018430464, 97754949812224, 1120674238611456, 12847530427547648, 147285426432966656, 1688495240694988800, 19357081676605554688, 221911554309549457408, 2544016621769302474752
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Keywords

Comments

Eighth binomial transform of A162466.

Links

Crossrefs

Cf. A162466, A161728, A153594.

Programs

  • Mathematica
    Join[{a=1,b=16},Table[c=16*b-52*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
LinearRecurrence[{16,-52},{1,16},20] (* Harvey P. Dale, Dec 23 2020 *)
  • PARI
    F=nfinit(x^2-3); for(n=0, 17, print1(nfeltdiv(F, ((4+x)*(8+2*x)^n-(4-x)*(8-2*x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009
  • PARI
    Vec(1/(1-16*x+52*x^2)+O(x^25)) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = 16*a(n-1) - 52(n-2) for n > 1; a(0) = 1, a(1) = 16.
G.f.: 1/(1 - 16*x + 52*x^2). - Klaus Brockhaus, Jun 19 2009
a(n) = 2^n*A153594(n). - M. F. Hasler, Dec 03 2014
E.g.f.: exp(8*x)*(3*cosh(2*sqrt(3)*x) + 4*sqrt(3)*sinh(2*sqrt(3)*x))/3. - Stefano Spezia, Dec 31 2022

Extensions

Extended beyond a(5) by Klaus Brockhaus, Jun 19 2009
Edited by Klaus Brockhaus, Jul 05 2009, and M. F. Hasler, Dec 03 2014

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 1-5 of 5 results.

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