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.

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

A161728 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n-(3-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).

Original entry on oeis.org

1, 7, 43, 253, 1465, 8431, 48403, 277621, 1591729, 9124759, 52305595, 299822893, 1718610409, 9851185663, 56467549987, 323674986277, 1855321740385, 10634799101479, 60959210186827, 349421293175389, 2002900612974361, 11480728092514831, 65808116771451955, 377215468968922837
Offset: 0

Views

Author

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

Keywords

Comments

Fourth binomial transform of A162436, inverse binomial transform of A162272.
The inverse binomial transform yields A030192. The binomial transform yields A162272. - R. J. Mathar, Jul 07 2009

Links

Crossrefs

Cf. A162436, A162272, A030192, A161729.

Programs

  • PARI
    F=nfinit(x^2-3); for(n=0, 20, print1(nfeltdiv(F, ((3+x)*(4+x)^n-(3-x)*(4-x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009
  • PARI
    Vec((1-x)/(1-8*x+13*x^2)+O(x^25)) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = 8*a(n-1) - 13(n-2) for n > 1; a(0) = 1, a(1) = 7.
G.f.: (1 - x)/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Jun 19 2009
E.g.f.: exp(4*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Extensions

Extended beyond a(5) by Klaus Brockhaus, Jun 19 2009
Edited by Klaus Brockhaus, Jul 05 2009; 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 1-3 of 3 results.

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