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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A090654 Decimal expansion of 4 + 2*sqrt(6).

Original entry on oeis.org

8, 8, 9, 8, 9, 7, 9, 4, 8, 5, 5, 6, 6, 3, 5, 6, 1, 9, 6, 3, 9, 4, 5, 6, 8, 1, 4, 9, 4, 1, 1, 7, 8, 2, 7, 8, 3, 9, 3, 1, 8, 9, 4, 9, 6, 1, 3, 1, 3, 3, 4, 0, 2, 5, 6, 8, 6, 5, 3, 8, 5, 1, 3, 4, 5, 0, 1, 9, 2, 0, 7, 5, 4, 9, 1, 4, 6, 3, 0, 0, 5, 3, 0, 7, 9, 7, 1, 8, 8, 6, 6, 2, 0, 9, 2, 8, 0, 4, 6, 9, 6
Offset: 1

Views

Author

Felix Tubiana, Feb 05 2004

Keywords

Comments

Equals n +n/(n +n/(n +n/(n +....))) for n = 8. See also A090388. - Stanislav Sykora, Jan 23 2014

Examples

			8.898979485566356196394568149...

Links

Crossrefs

Cf. A010480, A176213.
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Essentially the same as A010480.

Programs

  • Mathematica
    RealDigits[4 + 2*Sqrt[6], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)
  • PARI
    4 + 2*sqrt(6) \\ G. C. Greubel, Jul 03 2017

A268409 a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5.

Original entry on oeis.org

3, 5, 26, 114, 508, 2260, 10056, 44744, 199088, 885840, 3941536, 17537824, 78034368, 347213120, 1544921216, 6874111104, 30586286848, 136093369600, 605546052096, 2694370947584, 11988575894528, 53343045473280, 237349333682176, 1056083425675264
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

Comments

In general, the ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m, is (k - (k*r - m)*x)/(1 - r*x - s*x^2). This recurrence gives the closed form b(n) = (2^(-n - 1)*((k*r - 2*m)*(r - sqrt(r^2 + 4*s))^n + (2*m - k*r)*(sqrt(r^2 + 4*s) + r)^n + k*sqrt(r^2 + 4*s)*(r - sqrt(r^2 + 4*s))^n + k*sqrt(r^2 + 4*s)*(sqrt(r^2 + 4*s) + r)^n))/sqrt(r^2 + 4*s).

Links

Crossrefs

Cf. A021001, A084059, A090017, A107979, A164549, A176213, A189741.

Programs

  • Magma
    [n le 2 select 2*n+1 else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2016
  • Mathematica
    RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == 4 a[n - 1] + 2 a[n - 2]}, a, {n, 23}]
LinearRecurrence[{4, 2}, {3, 5}, 24]
Table[((18 + Sqrt[6]) (2 - Sqrt[6])^n - (Sqrt[6] - 18) (2 + Sqrt[6])^n)/12, {n, 0, 23}]
  • PARI
    Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

Formula

G.f.: (3 - 7*x)/(1 - 4*x - 2*x^2).
a(n) = ((18 + sqrt(6))*(2 - sqrt(6))^n - (sqrt(6) - 18)*(2 + sqrt(6))^n)/12.
Lim_{n -> infinity} a(n + 1)/a(n) = 2 + sqrt(6) = A176213.
a(n) = 3*A090017(n+1) -7*A090017(n). - R. J. Mathar, Mar 12 2017
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