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.

A014125 Bisection of A001400.

Original entry on oeis.org

1, 3, 6, 11, 18, 27, 39, 54, 72, 94, 120, 150, 185, 225, 270, 321, 378, 441, 511, 588, 672, 764, 864, 972, 1089, 1215, 1350, 1495, 1650, 1815, 1991, 2178, 2376, 2586, 2808, 3042, 3289, 3549, 3822, 4109, 4410, 4725, 5055, 5400, 5760, 6136, 6528, 6936, 7361, 7803
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also Schoenheim bound L_1(n,5,4).
Degrees of polynomials defined by p(n) = (x^(n+1)*p(n-1)p(n-3) + p(n-2)^2)/p(n-4), p(-4)=p(-3)=p(-2)=p(-1)=1. - Michael Somos, Jul 21 2004
Degrees of polynomial tau-functions of q-discrete Painlevé I, which generate sequence A095708 when q=2 (up to an offset of 3). - Andrew Hone, Jul 29 2004
Because of the Laurent phenomenon for the general q-discrete Painlevé I tau-function recurrence p(n) = (a*x^(n+1)*p(n-1)*p(n-3) + b*p(n-2)^2)/p(n-4), p(n) for n > -1 will always be a polynomial in x and a Laurent polynomial in a,b and the initial data p(-4),p(-3),p(-2),p(-1). - Andrew Hone, Jul 29 2004
Create the sequence 0,0,0,0,0,6,18,36,66,108,... so that the sum of three consecutive terms b(n) + b(n+1) + b(n+2) = A007531(n), with b(0)=0; then a(n) = b(n+5)/6. - J. M. Bergot, Jul 30 2013
Number of partitions of n into three kinds of part 1 and one kind of part 3. - Joerg Arndt, Sep 28 2015
First differences are A001840(k) starting with k=2; second differences are A086161(k) starting with k=1. - Bob Selcoe, Sep 28 2015
Maximum Wiener index of all maximal planar graphs with n+2 vertices. The extremal graphs are cubes of paths. - Allan Bickle, Jul 09 2022
Maximum Wiener index of all maximal 3-degenerate graphs with n+2 vertices. (A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices.) The extremal graphs are cubes of paths, so the bound also applies to 3-trees. - Allan Bickle, Sep 18 2022

Examples

			Polynomials: p(0)=x+1, p(1)=x^3+x^2+1, p(2)=x^6+x^5+x^3+x^2+2x+1, ...
a(12)=185:  A000217(13)=91 + a(9)=94 == 91+55+28+10+1 = 185. - _Bob Selcoe_, Sep 27 2015
a(3)=11: the 11 partitions of 3 are {1a,1a,1a}, {1a,1a,1b}, {1a,1a,1c}, {1a,1b,1b}, {1a,1b,1c}, {1a,1c,1c}, {1b,1b,1b}, {1b,1b,1c}, {1b,1c,1c}, {1c,1c,1c}, {3}. - _Bob Selcoe_, Oct 04 2015

References

  • W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992. See Eq. 1.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • L. Smiley, Hidden Hexagons (preprint).

Links

Crossrefs

Cf. A000217, A000631, A001840, A014126, A086161, A095708.
A column of A036838.
Cf. A002623, A014125, A122046, A122047. - Johannes W. Meijer, May 20 2011
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, A175724.

Programs

  • Magma
    [n^3/18+n^2/2+4*n/3+1+(((n+1) mod 3)-1)/9 : n in [0..50]]; // Wesley Ivan Hurt, Apr 14 2015
  • Magma
    I:=[1,3,6,11,18,27]; [n le 6 select I[n] else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3)-3*Self(n-4)+3*Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Apr 15 2015
  • Maple
    L := proc(v,k,t,l) local i,t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v,k,t). Current sequence is L_1(n,n-3,n-4,1).
  • Mathematica
    CoefficientList[Series[1/((1 - x)^3*(1 - x^3)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 14 2015 *)
  • PARI
    a(n)=if(n<-5,-a(-6-n),polcoeff(1/(1-x)^3/(1-x^3)+x^n*O(x),n)) /* Michael Somos, Jul 21 2004 */
  • PARI
    my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^3))) \\ Altug Alkan, Oct 16 2015
  • PARI
    a(n)=(n^3 + 9*n^2 + 24*n + 19)\/18 \\ Charles R Greathouse IV, Jun 29 2020
  • Sage
    [(binomial(n+4,3) - ((n+4)//3))/3 for n in (0..50)] # G. C. Greubel, Apr 28 2019

Formula

G.f.: 1/((1-x)^3*(1-x^3)).
a(n) = -a(-6-n) = 3*a(n-1) -3*a(n-2) +2*a(n-3) -3*a(n-4) +3*a(n-5) -a(n-6).
The simplest recurrence is fourth order: a(n) = a(n-1) + a(n-3) - a(n-4) + n + 1, which gives the g.f.: 1/((1-x)^3*(1-x^3)), with a(-4) = a(-3) = a(-2) = a(-1) = 0.
a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + 2/(9*sqrt(3))*sin(2*Pi*n/3). - Andrew Hone, Jul 29 2004
a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + (((n+1) mod 3) - 1)/9. - same formula, simplified by Gerald Hillier, Apr 14 2015
a(n) = (2*A000027(n+1) + 3*A000292(n+1) + A049347(n-1) + 1 + 3*A000217(n+1))/9. - R. J. Mathar, Nov 16 2007
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A144677(n) + A144677(n-1) + A144677(n-2).
a(n) = A190717(n-4) + 2*A190717(n-3) + 3*A190717(n-2) + 2*A190717(n-1) + A190717(n). (End)
3*a(n) = binomial(n+4,3) - floor((n+4)/3). - Bruno Berselli, Nov 08 2013
a(n) = A000217(n+1) + a(n-3) = Sum_{j>=0, n>=3*j} (n-3*j+1)*(n-3*j+2)/2. - Bob Selcoe, Sep 27 2015
a(n) = round(((2*n+5)^3 + 3*(2*n+5)^2 - 9*(2*n+5))/144). - Giacomo Guglieri, Jun 28 2020
a(n) = floor(((n+2)^3 + 3*(n+2)^2)/18). - Allan Bickle, Aug 01 2020
a(n) = Sum_{j=0..n} (n-j+1)*floor((j+3)/3). - G. C. Greubel, Oct 18 2021
E.g.f.: exp(x) + exp(x)*x*(34 + 12*x + x^2)/18 + 2*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - Stefano Spezia, Apr 05 2023

Extensions

More terms from James Sellers, Dec 24 1999

A086162 Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^3.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 5, 7, 8, 8, 11, 11, 11, 14, 15, 15, 19, 19, 19, 23, 24, 24, 29, 29, 29, 34, 35, 35, 41, 41, 41, 47, 48, 48, 55, 55, 55, 62, 63, 63, 71, 71, 71, 79, 80, 80, 89, 89, 89, 98, 99, 99, 109, 109, 109, 119, 120, 120, 131, 131, 131, 142, 143, 143, 155
Offset: 0

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Author

Jan Snellman, Aug 25 2003

Keywords

Comments

Alternatively, "concave partitions" of n with at most 3 parts, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do not lie in the Ferrers diagram of the partition, is integrally closed.

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
  • M. Paulsen and J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.

Links

Crossrefs

Cf. A084913, A086161, A086163.

Programs

  • Maple
    f:= gfun:-rectoproc({a(i+10)=a(i)-a(i+1)-a(i+3)+a(i+4)-a(i+6)+a(i+7)+a(i+9), seq(a(i)=[1, 1, 2, 3, 3, 5, 5, 5, 7, 8][i+1],i=0..9)},a(i),remember):
map(f, [$0..100]); # Robert Israel, May 22 2015
  • Mathematica
    LinearRecurrence[{1, 0, 1, -1, 0, 1, -1, 0, -1, 1}, {1, 1, 2, 3, 3, 5, 5, 5, 7, 8}, 60] (* Jean-François Alcover, Aug 16 2022 *)
  • PARI
    Vec((1+t^2+t^5-2*t^6-t^8+t^9)/((1-t)*(1-t^3)*(1-t^6)) + O(t^80)) \\ Michel Marcus, May 22 2015

Formula

G.f.: (1+t^2+t^5-2*t^6-t^8+t^9)/((1-t)*(1-t^3)*(1-t^6)).

A084913 Number of monomial ideals in two variables that are Artinian, integrally closed and of colength n.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 11, 17, 23, 28, 39, 48, 59, 79, 100, 121, 152, 185, 225, 280, 338, 404, 492, 584, 696, 835, 983, 1162, 1385, 1612
Offset: 0

Views

Author

Jan Snellman and Michael Paulsen, Jul 03 2003

Keywords

Comments

Alternatively, "concave partitions" of n, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do not lie in the Ferrers diagram of the partition, is integrally closed.

Examples

			a(4) = 4 because the Artinian monomial ideals in two variables that have colength 4 are (x^4,y), (x^3,y^2), (x^2, y^2), (x^2,xy,y^3), (x,y^4), corresponding to the partitions (1,1,1,1), (3,1), (2,2), (2,1,1), (4); the ideal (x^2,y^2) is not integrally closed, hence the partition (2,2) is not concave.

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
  • M. Paulsen & J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.

Links

Crossrefs

Cf. A086161, A086162, A086163.
Showing 1-3 of 3 results.

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