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.

A236773 a(n) = n + floor( n^2/2 + n^3/3 ).

Original entry on oeis.org

0, 1, 6, 16, 33, 59, 96, 145, 210, 292, 393, 515, 660, 829, 1026, 1252, 1509, 1799, 2124, 2485, 2886, 3328, 3813, 4343, 4920, 5545, 6222, 6952, 7737, 8579, 9480, 10441, 11466, 12556, 13713, 14939, 16236, 17605, 19050, 20572, 22173, 23855, 25620, 27469
Offset: 0

Views

Author

Bruno Berselli, Feb 07 2014

Keywords

Comments

This sequence follows A074148 and A042965, A236771.
The prime terms are 59, 829, 14939, 35759, 93719, 132409, 155219, 290399, 414179, 487463, ... .
If a(k) is prime then k == 1, 5, 7 or 11 (mod 12).
Third differences: 1, 2, 2, 2, 1, 4 repeated (unsigned terms of A181982).
Fourth differences: 1, 0, 0, -1, 3, -3 repeated (see A131193).

Crossrefs

Cf. A074148: n+floor(n^2/2).
Cf. A042965: n+floor(1/2+n/3); A236771: n+floor(n/2+n^2/3).
Cf. A236772: floor(sum(i=1..n, n^i/i)).

Programs

  • Magma
    [n+Floor(n^2/2+n^3/3): n in [0..50]];
    
  • Magma
    I:=[0,1,6,16,33,59,96,145,210]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-6)-3*Self(n-7)+3*Self(n-8)-Self(n-9): n in [1..50]]; // Vincenzo Librandi, Feb 08 2014
    
  • Maple
    seq(n+floor(n^2/2+n^3/3),n=0..43); # Paolo P. Lava, Aug 24 2018
  • Mathematica
    Table[n + Floor[n^2/2 + n^3/3], {n, 0, 50}]
    CoefficientList[Series[x (1 + 3 x + x^2 + 2 x^3 + 2 x^4 + 2 x^5 + x^7)/((1 + x) (1 - x + x^2) (1 + x + x^2) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 08 2014 *)
  • PARI
    vector(60, n, n--; n+floor(n^2/2 +n^3/3)) \\ G. C. Greubel, Aug 12 2018

Formula

G.f.: x*(1+3*x+x^2+2*x^3+2*x^4+2*x^5+x^7) / ((1+x)*(1-x+x^2)*(1+x+x^2)*(1-x)^4).
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-6) -3*a(n-7) +3*a(n-8) -a(n-9).
Also, for h>=0:
a(6h) = 6*h*( 12*h^2 + 3*h + 1 ),
a(6h+1) = 72*h^3 + 54*h^2 + 18*h + 1,
a(6h+2) = 6*( 4*h + 1 )*( 3*h^2 + 3*h + 1 ),
a(6h+3) = 2*( 36*h^3 + 63*h^2 + 39*h + 8 ),
a(6h+4) = 3*( 24*h^3 + 54*h^2 + 42*h + 11 ),
a(6h+5) = 72*h^3 + 198*h^2 + 186*h + 59.

A237872 Numerator of Sum_{i=1..n} n^i/i.

Original entry on oeis.org

1, 4, 33, 292, 10085, 48756, 2827293, 257063528, 13779684369, 70889442280, 72634140523901, 314690934778068, 140915129117772841, 5533416685634616884, 251767541303505518145, 55644156684309383260624, 7481965178603932789388755
Offset: 1

Views

Author

Bruno Berselli, Feb 14 2014

Keywords

Comments

The sequence gives the numerators of -n^(n+1)*Phi(n,1,n+1)-log(-n+1) for n>1, where Phi is the Lerch transcendent.

Crossrefs

Cf. A236772, A237873 (denominators).

Programs

  • Magma
    terms:=20; s:=[&+[n^i/i: i in [1..n]]: n in [1..terms]]; [Numerator(s[n]): n in [1..terms]];
  • Maple
    A237872:=n->numer(add(n^i/i, i=1..n)): seq(A237872(n), n=1..20); # Wesley Ivan Hurt, Apr 26 2017
  • Mathematica
    f[n_] := Sum[n^i/i, {i, 1, n}]; Table[Numerator[f[n]], {n, 1, 20}]

A237873 Denominator of Sum_{i=1..n} n^i/i.

Original entry on oeis.org

1, 1, 2, 3, 12, 5, 20, 105, 280, 63, 2520, 385, 5544, 6435, 8008, 45045, 144144, 85085, 116688, 2909907, 739024, 146965, 232792560, 37182145, 356948592, 128707425, 2974571600, 717084225, 80313433200, 215656441, 30247916400, 4512611027925, 486207248800
Offset: 1

Views

Author

Bruno Berselli, Feb 14 2014

Keywords

Comments

The sequence gives the denominators of -n^(n+1)*Phi(n,1,n+1)-log(-n+1) for n>1, where Phi is the Lerch transcendent.

Crossrefs

Cf. A236772, A237872 (numerators)

Programs

  • Magma
    terms:=40; s:=[&+[n^i/i: i in [1..n]]: n in [1..terms]]; [Denominator(s[n]): n in [1..terms]];
  • Mathematica
    f[n_] := Sum[n^i/i, {i, 1, n}]; Table[Denominator[f[n]], {n, 1, 40}]
Showing 1-3 of 3 results.