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.

A324467 Three times the binary weight of n: 3*A000120(n).

Original entry on oeis.org

0, 3, 3, 6, 3, 6, 6, 9, 3, 6, 6, 9, 6, 9, 9, 12, 3, 6, 6, 9, 6, 9, 9, 12, 6, 9, 9, 12, 9, 12, 12, 15, 3, 6, 6, 9, 6, 9, 9, 12, 6, 9, 9, 12, 9, 12, 12, 15, 6, 9, 9, 12, 9, 12, 12, 15, 9, 12, 12, 15, 12, 15, 15, 18, 3, 6, 6, 9, 6, 9, 9, 12, 6, 9, 9, 12, 9, 12, 12, 15, 6, 9, 9, 12
Offset: 0

Views

Author

N. J. A. Sloane, Mar 02 2019

Keywords

Comments

Exponent of highest power of 2 that divides multinomial(4*n;n,n,n,n).

Links

Crossrefs

Cf. A000120, A006480, A324465, A324466.

Programs

  • Magma
    [3*Valuation(Binomial(2*n,n), 2): n in [0..120]]; // Vincenzo Librandi, Mar 11 2019
  • Maple
    [seq(padic[ordp](multinomial(4*n,n,n,n,n),2),n=0..128)];
  • Mathematica
    3 Log[2, Nest[Join[#, 2 #]&, {1}, 8]] (* Vincenzo Librandi, Mar 11 2019 *)

A324152 a(0)=1; for n>0, a(n) = (3/((n+1)*(n+2)*(n+3))) * multinomial(4*n;n,n,n,n).

Original entry on oeis.org

1, 3, 126, 9240, 900900, 104756652, 13742520792, 1968826448160, 301700280152700, 48756255150603000, 8226155369009738160, 1438285479229504301760, 259131100507849025033760, 47897087290614993606462000, 9050997011303368719799740000
Offset: 0

Views

Author

Michael De Vlieger and N. J. A. Sloane, Mar 01 2019

Keywords

Comments

It is conjectured that a(n) is always an integer.
If all terms except the first are doubled, we get A324478, which IS known to be integral.

Links

Crossrefs

Cf. A000108, A324151, A324465 (exponent of 2), A324467, A324478.

Programs

  • Magma
    [1] cat [n le 1 select 3 else Self(n-1)*4*(4*n-3)*(4*n-2)*(4*n-1)/((n)^2*(n+3)): n in [1..20]]; // Vincenzo Librandi, Mar 11 2019
  • Mathematica
    c[m_, n_] := m Product[1/(n + i), {i, m}] (Multinomial @@ ConstantArray[n, m + 1]); {1}~Join~Array[c[3, #] &, 14] (* Michael De Vlieger, Mar 01 2019 *)
Flatten[{1, Table[3*(4*n)! / ((n!)^3 * (n+3)!), {n, 1, 15}]}] (* Vaclav Kotesovec, Jul 21 2019 *)

Formula

a(n+1) = a(n)*4*(4*n+1)*(4*n+2)*(4*n+3)/((n+1)^2*(n+4)) for n>0.
From Vaclav Kotesovec, Jul 21 2019: (Start)
For n>0, a(n) = 3*(4*n)! / ((n!)^3 * (n+3)!).
a(n) ~ 3 * 2^(8*n - 1/2) / (Pi^(3/2) * n^(9/2)). (End)

A324468 a(n) = r(n) + r(n+1) + r(n+2), where r(n) is the ruler sequence A007814.

Original entry on oeis.org

1, 3, 2, 3, 1, 4, 3, 4, 1, 3, 2, 3, 1, 5, 4, 5, 1, 3, 2, 3, 1, 4, 3, 4, 1, 3, 2, 3, 1, 6, 5, 6, 1, 3, 2, 3, 1, 4, 3, 4, 1, 3, 2, 3, 1, 5, 4, 5, 1, 3, 2, 3, 1, 4, 3, 4, 1, 3, 2, 3, 1, 7, 6, 7, 1, 3, 2, 3, 1, 4, 3, 4, 1, 3, 2, 3, 1, 5, 4, 5, 1, 3, 2, 3, 1
Offset: 1

Views

Author

N. J. A. Sloane, Mar 03 2019

Keywords

Links

Crossrefs

Cf. A001511, A007814, A050603 (r(n)+r(n+1)), A324465.

Programs

  • Magma
    [&+[Valuation(n+k, 2): k in [0..2]]: n in [1..70]]; // Vincenzo Librandi, Mar 10 2019
  • Mathematica
    Table[Sum[IntegerExponent[n + k, 2], {k, 0, 2}], {n, 100}] (* Vincenzo Librandi, Mar 10 2019 *)
  • PARI
    a(n) = sum(k=0, 2, valuation(n+k, 2)); \\ Michel Marcus, Mar 10 2019
  • Python
    def A324468(n): return (~n & n-1).bit_length()+(~(n+1) & n).bit_length()+(~(n+2) & n+1).bit_length() # Chai Wah Wu, Jul 01 2022

Formula

1 <= a(n) <= 1 + log_2(n+2). - Charles R Greathouse IV, Jul 01 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. - Amiram Eldar, Sep 10 2024
Showing 1-3 of 3 results.

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

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