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.

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A218987 Power ceiling sequence of 2+sqrt(7).

Original entry on oeis.org

5, 24, 112, 521, 2421, 11248, 52256, 242769, 1127845, 5239688, 24342288, 113088217, 525379733, 2440783584, 11339273536, 52679444897, 244735600197, 1136980735480, 5282129742512, 24539461176489, 114004233933493, 529635319263440, 2460553978854240
Offset: 0

Views

Author

Clark Kimberling, Nov 11 2012

Keywords

Comments

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(7), and the limit p4(r) = 5.19758760498048832156707270895307875397561324042...
See A218986 for the power floor function, p1(x); for comparison of p1 and p4, limit(p4(r)/p1(r) = 4 - sqrt(7).

Examples

			a(0) = ceiling(r) = 5, where r = 2+sqrt(7);
a(1) = ceiling(5*r) = 24; a(2) = ceiling(24*r) = 112.

Links

Crossrefs

Cf. A214992, A015530, A126473, A218986.

Programs

  • Mathematica
    (See A218986.)
  • PARI
    a(n) = round((-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84) \\ Colin Barker, Sep 02 2016
  • PARI
    Vec((5-x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ Colin Barker, Sep 02 2016

Formula

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(7), a(0) = ceiling(x).
a(n) = 5*a(n-1) - a(n-2) - 3*a(n-3).
G.f.: (5 - x - 3*x^2)/(1 - 5*x + x^2 + 3*x^3).
a(n) = (-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84. - Colin Barker, Sep 02 2016
E.g.f.: exp(x)*(exp(x)*(217*cosh(sqrt(7)*x) + 83*sqrt(7)*sinh(sqrt(7)*x)) - 7)/42. - Stefano Spezia, Aug 05 2025

A239549 Expansion of x/(1-8*x-12*x^2).

Original entry on oeis.org

0, 1, 8, 76, 704, 6544, 60800, 564928, 5249024, 48771328, 453158912, 4210527232, 39122124800, 363503325184, 3377492099072, 31381976694784, 291585718747136, 2709269470314496, 25173184387481600, 233896708743626752, 2173251882598793216
Offset: 0

Views

Author

Felix P. Muga II, Mar 21 2014

Keywords

Comments

The limit of a(n+1)/a(n) is equal to 2+sqrt(7) as n approaches infinity.
This is the Lucas sequence U(8,-12).
For any three-term recurrence S(n) = S(n-1)*x1 + S(n-2)*x2*1, with S(-1) = 0 and S(0) = 1, with n-independent coefficients (like here x1=8 and x2=12) one can use the standard Morse code with a dot of length 1 standing for x1 and a dash of length 2 standing for x2. The Morse code polynomial S(x1,x2;n) is then obtained by summing over all codes of length n. E.g., S(x1,x2;3) = x1^3 + 2*x1*x2 from dot dot dot, dot dash and dash dot. Here x1=8 and x2=12 (labeled dots and dashes). For example, S(3) = 8*(8^2 + 2*12) = 704 = b(3) = a(4), because in a the offset differs from the one for S. See the Graham et al. book, on Morse code polynomials (Euler's continuants), p 302. This comment was motivated by an earlier one from the author of this sequence. - Wolfdieter Lang, Mar 27 2014
a(n-1) (for n>=1) is the number of compositions of n into 8 kinds of parts 1 and 12 kinds of parts 2. - Joerg Arndt, Mar 26 2014

References

  • R. L. Graham, D. E. Knuth, L. O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.

Links

Crossrefs

Cf. A000079, A011782, A015530.

Programs

  • Haskell
    a239549 n = a239549_list !! n
a239549_list = 0 : 1 : zipWith (+)
               (map (* 8) $ tail a239549_list) (map (* 12) a239549_list)
-- Reinhard Zumkeller, Feb 20 2015
  • Mathematica
    CoefficientList[Series[x / (1 - 8 x - 12 x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 28 2014 *)
LinearRecurrence[{8,12},{0,1},30] (* Harvey P. Dale, Nov 14 2021 *)

Formula

a(n) = 8*a(n-1) + 12*a(n-2) for n > 1 and a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x - 12*x^2).
a(n) = (1/(4*sqrt(7)))*( (4+2*sqrt(7))^n - (4-2*sqrt(7))^n ).
a(0) = 0 and a(n) = 2^(n-1) * A015530(n) for n > 0.
a(n) = A011782(n) * A015530(n) =2^(n-1)*A015530(n) for n >= 0.
a(n+1) = b(n) = sum(binomial(n-k, k)*8^(n-2*k)*12^k, k = 0..floor(n/2)), n>=0, b(-1) := 0. From Morse code counting, with n-2*k the number of dots and k the number of dashes for code length n. See the comment and example for b(3) = S(3) above. - Wolfdieter Lang, Mar 26 2014

A106570 a(n) = 4*a(n-1) + 3*a(n-2), with a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 12, 57, 264, 1227, 5700, 26481, 123024, 571539, 2655228, 12335529, 57307800, 266237787, 1236874548, 5746211553, 26695469856, 124020514083, 576168465900, 2676735405849, 12435447021096, 57771994301931, 268394318271012
Offset: 0

Views

Author

Roger L. Bagula, May 30 2005

Keywords

Links

Crossrefs

Cf. A015530, A106569.

Programs

  • Magma
    [n le 2 select 3*(n-1) else 4*Self(n-1) + 3*Self(n-2): n in [1..41]]; // G. C. Greubel, Sep 07 2021
  • Maple
    a[0]:=0: a[1]:=3: for n from 2 to 24 do a[n]:=4*a[n-1]+3*a[n-2] od: seq(a[n], n=0..40);
  • Mathematica
    LinearRecurrence[{4,3}, {0,3}, 41] (* G. C. Greubel, Sep 07 2021 *)
  • Sage
    [3*lucas_number1(n,4,-3) for n in (0..40)] # G. C. Greubel, Sep 07 2021

Formula

G.f.: 3*x/(1 - 4*x - 3*x^2).
a(n) = 4*a(n-1) + 3*a(n-2), a(0)=0, a(1)=3.
a(n) = 3*A015530(n).

Extensions

Edited by N. J. A. Sloane, Apr 30 2006
New name by G. C. Greubel, Sep 07 2021
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