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.

A153597 a(n) = ((6 + sqrt(3))^n - (6 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 12, 111, 936, 7569, 59940, 469503, 3656016, 28378593, 219894588, 1702241487, 13170376440, 101870548209, 787824155988, 6092161780959, 47107744223904, 364251591915201, 2816463543593580, 21777259989921327, 168383822940467784
Offset: 1

Views

Author

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

Keywords

Comments

Fourth binomial transform of A055845.
lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(3) = 7.73205080756887729....

Links

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A055845.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
  • Magma
    I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-33*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016
  • Mathematica
    LinearRecurrence[{12,-33},{1, 12},25] (* G. C. Greubel, Aug 22 2016 *)
  • Sage
    [lucas_number1(n,12,33) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

Formula

G.f.: x/(1 - 12*x + 33*x^2). - Klaus Brockhaus, Dec 31 2008, (corrected Oct 11 2009)
a(n) = 12*a(n-1) - 33*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(6*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

A166450 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 3, 18, 9, 54, 27, 162, 81, 486, 243, 1458, 729, 4374, 2187, 13122, 6561, 39366, 19683, 118098, 59049, 354294, 177147, 1062882, 531441, 3188646, 1594323, 9565938, 4782969, 28697814, 14348907, 86093442, 43046721, 258280326
Offset: 1

Views

Author

Klaus Brockhaus, Oct 13 2009

Keywords

Comments

Interleaving of A000244 and 6*A000244.
Second binomial transform is A055845. Sixth binomial transform is A153597.

Links

Crossrefs

Cf. A000244 (powers of 3), A055845, A153597.

Programs

  • Magma
    [ n le 2 select 5*n-4 else 3*Self(n-2): n in [1..35] ];
  • Mathematica
    LinearRecurrence[{0,3},{1,6}, 50] (* G. C. Greubel, May 14 2016 *)

Formula

a(n) = (3+(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(1+6*x)/(1-3*x^2).

A257487 Expansion of ( -4+15*x-8*x^2 ) / ( (x-1)*(x^2-4*x+1) ).

Original entry on oeis.org

4, 5, 13, 44, 160, 593, 2209, 8240, 30748, 114749, 428245, 1598228, 5964664, 22260425, 83077033, 310047704, 1157113780, 4318407413, 16116515869, 60147656060, 224474108368, 837748777409, 3126521001265, 11668335227648, 43546819909324
Offset: 0

Views

Author

R. J. Mathar, Apr 26 2015

Keywords

Links

Crossrefs

Cf. A055845 (first differences).

Programs

  • Maple
    A257487 := proc(n)
    (5+sqrt(3))/4*(2-sqrt(3))^n+(5-sqrt(3))/4*(2+sqrt(3))^n+3/2 ;
    expand(%) ;
end proc:
seq(A257487(n),n=0..30) ;
  • Mathematica
    CoefficientList[Series[(-4+15x-8x^2)/((x-1)(x^2-4x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{5,-5,1},{4,5,13},30] (* Harvey P. Dale, Dec 29 2024 *)
  • PARI
    Vec(( -4+15*x-8*x^2 ) / ( (x-1)*(x^2-4*x+1) ) + O(x^50)) \\ Michel Marcus, Apr 26 2015

Formula

a(n) = (5*A001353(n+1)-13*A001353(n)+3)/2. - R. J. Mathar, May 26 2016
Showing 1-3 of 3 results.

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