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.

A040072 Continued fraction for sqrt(82).

Original entry on oeis.org

9, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			9.05538513813741662657380... = 9 + 1/(18 + 1/(18 + 1/(18 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 10 2009

Links

Crossrefs

Cf. A010533 (decimal expansion), A041144/A041145 (convergents), A248305 (Egyptian fraction).
Cf. A040000, A040006.

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[82],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
PadRight[{9},120,{18}] (* Harvey P. Dale, Oct 09 2020 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 51000); x=contfrac(sqrt(82)); for (n=0, 20000, write("b040072.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 10 2009

Formula

From Elmo R. Oliveira, Feb 10 2024: (Start)
a(n) = 18 = A010857(n) for n >= 1.
G.f.: 9*(1+x)/(1-x).
E.g.f.: 18*exp(x) - 9.
a(n) = 9*A040000(n) = 3*A040006(n). (End)

A023016 Number of partitions of n into parts of 18 kinds.

Original entry on oeis.org

1, 18, 189, 1482, 9576, 53676, 269325, 1235286, 5256711, 20985272, 79260723, 285139764, 982349361, 3255488082, 10416507579, 32281134120, 97154549289, 284625019800, 813310723925, 2270826800172, 6204926551824, 16615751700618
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

a(n) is Euler transform of A010857. - Alois P. Heinz, Oct 17 2008

Links

Crossrefs

Cf. 18th column of A144064. - Alois P. Heinz, Oct 17 2008

Programs

  • Maple
    with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*18, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008
  • Mathematica
    CoefficientList[Series[1/QPochhammer[x]^18, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)
  • PARI
    Vec(1/eta(x)^18 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

a(0) = 1, a(n) = (18/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
a(n) ~ m^((m+1)/4) * exp(Pi*sqrt(2*m*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)) * (1 - ((9+Pi^2)*m^2+36*m+27) / (24*Pi*sqrt(6*m*n))), set m = 18. - Vaclav Kotesovec, Jun 28 2025

A249693 a(4n) = 3*n+1, a(2n+1) = 3*n+2, a(4n+2) = 3*n.

Original entry on oeis.org

1, 2, 0, 5, 4, 8, 3, 11, 7, 14, 6, 17, 10, 20, 9, 23, 13, 26, 12, 29, 16, 32, 15, 35, 19, 38, 18, 41, 22, 44, 21, 47, 25, 50, 24, 53, 28, 56, 27, 59, 31, 62, 30, 65, 34, 68, 33, 71, 37, 74, 36, 77, 40, 80, 39, 83, 43, 86, 42, 89, 46, 92, 45
Offset: 0

Views

Author

Paul Curtz, Dec 03 2014

Keywords

Comments

A permutation of the nonnegative numbers.

Links

Crossrefs

Cf. A001477, A010704, A010857, A016789, A061037.

Programs

  • Magma
    m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 2*x - x^2 + 3*x^3 + 3*x^4 + x^5)/(1 - x^2 - x^4 + x^6))); // G. C. Greubel, Sep 20 2018
  • Mathematica
    a[n_] := (1/8)*(3*(-1)^(n+1)*(n+1)+9*n+10*{1, 0, -1, 0}[[Mod[n, 4]+1]]+1); Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 04 2014, after Robert Israel *)
  • PARI
    x='x+O('x^75); Vec((1 + 2*x - x^2 + 3*x^3 + 3*x^4 + x^5)/(1 - x^2 - x^4 + x^6)) \\ G. C. Greubel, Sep 20 2018

Formula

a(n+4) = a(n) + (sequence of period 2: repeat 3, 6).
a(4n+1) = 2*a(4n).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) is the rank of A061037(n) = -1, -3, 0, 5, ... in A247829(n) = 0, -1, -3, 2, ... .
G.f.: (1 + 2*x - x^2 + 3*x^3 + 3*x^4 + x^5)/(1 - x^2 - x^4 + x^6).
a(n) = (1 + 9*n - 3*(n+1)*(-1)^n + 10*cos(n*Pi/2))/8. - Robert Israel, Dec 03 2014
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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