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.

A134841 Triangle read by rows: first n terms of n-th row of an array formed by A126988 * A053731(transform).

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 6, 4, 7, 5, 5, 5, 5, 6, 6, 9, 8, 9, 6, 12, 7, 7, 7, 7, 7, 7, 8, 8, 12, 8, 14, 8, 12, 8, 15, 9, 9, 12, 9, 9, 12, 9, 9, 13, 10, 15, 10, 15, 12, 15, 10, 15, 10, 18
Offset: 0

Views

Author

Gary W. Adamson, Nov 12 2007

Keywords

Comments

Right border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).
Row sums = A001157: (1, 5, 10, 21, 26, 50, 50, ...).

Examples

			First few terms of the array:
  1,  1,  1,  1,  1,  1,  1, ...
  2,  3,  2,  3,  2,  3,  2, ...
  3,  3,  4,  3,  3,  4,  3, ...
  4,  6,  4,  7,  4,  6,  4, ...
  5,  5,  5,  5,  5,  6,  5, ...
  6,  9,  8,  9,  6, 12,  6, ...
  7,  7,  7,  7,  7,  7,  8, ...
  ...
First few rows of the triangle:
  1;
  2,  3;
  3,  3,  4;
  4,  6,  4,  7;
  5,  5,  5,  5,  6;
  6,  9,  8,  9,  6, 12;
  7,  7,  7,  7,  7,  7,  8,
  ...
		

Crossrefs

A053618 a(n) = ceiling(binomial(n,4)/n).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 9, 14, 21, 30, 42, 55, 72, 91, 114, 140, 170, 204, 243, 285, 333, 385, 443, 506, 575, 650, 732, 819, 914, 1015, 1124, 1240, 1364, 1496, 1637, 1785, 1943, 2109, 2285, 2470, 2665, 2870, 3086, 3311, 3548, 3795, 4054, 4324
Offset: 1

Views

Author

N. J. A. Sloane, Mar 25 2000

Keywords

Crossrefs

Programs

  • Magma
    [Ceiling(Binomial(n,4)/n): n in [1..60]]; // G. C. Greubel, May 16 2019
    
  • Mathematica
    CoefficientList[Series[x^4*(1-x+x^2)*(1-x+x^2+x^4)/((1-x)^3*(1-x^8)), {x,0,60}], x] (* G. C. Greubel, May 16 2019 *)
  • PARI
    concat([0,0,0], Vec(x^4*(x^2-x+1)*(x^4+x^2-x+1) / ((x-1)^4*(x+1)*(x^2+1)*(x^4+1)) + O(x^60))) \\ Colin Barker, Jan 20 2015
    
  • Sage
    [ceil(binomial(n,4)/n) for n in (1..60)] # G. C. Greubel, May 16 2019

Formula

a(n) = ( 2*n^3 - 12*n^2 + 22*n - 3 + 9*(-1)^n + 3*(1+(-1)^n)*(-1)^(n*(n-1)/2) - 6*(1 + (-1)^n)*(-1)^floor(n/4) )/48. - Luce ETIENNE, Jan 20 2015
G.f.: x^4*(1 - x + x^2)*(1 - x + x^2 + x^4)/((1-x)^3*(1-x^8)). - Colin Barker, Jan 20 2015

A053733 a(n) = ceiling(binomial(n,9)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 19, 55, 143, 334, 715, 1430, 2702, 4862, 8398, 13997, 22610, 35530, 54480, 81719, 120175, 173587, 246675, 345345, 476905, 650325, 876525, 1168700, 1542684, 2017356, 2615092, 3362260, 4289780, 5433722
Offset: 1

Views

Author

N. J. A. Sloane, Mar 25 2000

Keywords

Crossrefs

Cf. Sequences of the form ceiling(binomial(n,k)/n): A000012 (k=1), A004526 (k=2), A007997 (k=3), A008646 (k=5), A032192 (k=7), A053618 (k=4), A053643 (k=6), A053731 (k=8), this sequence (k=9).

Programs

  • Magma
    [Ceiling(Binomial(n,9)/n): n in [1..40]]; // G. C. Greubel, Sep 06 2019
    
  • Maple
    seq(ceil(binomial(n,9)/n), n=1..40); # G. C. Greubel, Sep 06 2019
  • Mathematica
    Table[Ceiling[Binomial[n, 9]/n], {n, 40}] (* G. C. Greubel, Sep 06 2019 *)
  • PARI
    vector(40, n, ceil(binomial(n,9)/n)) \\ G. C. Greubel, Sep 06 2019
    
  • Sage
    [ceil(binomial(n,9)/n) for n in (1..40)] # G. C. Greubel, Sep 06 2019
Showing 1-3 of 3 results.