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.

User: Brian G. Kronenthal

Brian G. Kronenthal's wiki page.

Brian G. Kronenthal has authored 4 sequences.

A346448 Number of nontrivial disconnected induced K_{1,3}-saturated graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 88, 620
Offset: 0

Keywords

Comments

A graph G is induced H-saturated if there exists no induced subgraph H in G, but for every edge e in the complement of G, the graph G+e has an induced subgraph H. We consider the complete graph K_n to be the trivial case. This is an empirical observation verified via a computer program.
Note that for n=11, the runtime is about 4 days.

A336764 Maximum number of order 3 subsquares in a Latin square of order n.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 7
Offset: 1

Keywords

Comments

A subsquare of a Latin square is a submatrix (not necessarily consisting of adjacent entries) which is itself a Latin square. (I. M. Wanless, Latin Squares with One Subsquare, Wiley and Sons)

Crossrefs

Formula

a(3^n) = 9*a(3^(n-1)) + 27^(n-1) (conjectured).

A246800 Even-indexed terms of A247984, a sequence motivated by generalized quadrangles.

Original entry on oeis.org

6, 10, 84, 186, 1276, 3172, 19816, 52666, 310764, 863820, 4899736, 14073060, 77509464, 228318856, 1228859344, 3693886906, 19513475404, 59644341436, 310223170744, 961665098956, 4936304385544, 15488087080696, 78602174905264, 249227373027556, 1252310513280376, 4007681094422392, 19961337935130096, 64408903437167496, 318297642651252784, 1034656923041985552
Offset: 1

Author

Brian G. Kronenthal, Nov 15 2014

Keywords

Comments

Let p be an odd prime, let e be a positive integer, and let q = p^e. Dmytrenko, Lazebnik, and Williford (2007) proved that every monomial graph of girth at least eight is isomorphic to G = G_q(xy, x^ky^(2k)) for some integer k which is not divisible by p. If q = 3, then G is isomorphic to G_3(xy, xy^2). If q >= 5, then F(x) = ((x + 1)^(2k) - 1)x^(q - 1 - k) - 2x^(q - 1) is a permutation polynomial, in which case the Hermite-Dickson Criterion implies that the coefficient at x^(q - 1) in F(x)^n must equal 0 modulo p. Term b(n) of sequence A247984 lists the constant term of the coefficient at x^(q - 1) in F(x)^n, and was first stated in Kronenthal (2012). The formula is defined piecewise, with b(n) = 2^n when n is odd and b(n) = 2^n - (-1)^(n/2)*binomial(n, n/2) when n is even. The sequence a(n) listed here consists of the even-indexed terms of A247984; in other words, a(n) = 2^(2n) - (-1)^(n)*binomial(2n, n). The provided Mathematica program produces the first 30 terms of the sequence.

Crossrefs

Equals even-indexed terms of A247984.

Programs

  • Magma
    [2^(2*n)-(-1)^n*Binomial(2*n, n) : n in [1..30]]; // Wesley Ivan Hurt, Nov 15 2014
  • Maple
    A246800:=n->2^(2*n)-(-1)^n*binomial(2*n, n): seq(A246800(n), n=1..30); # Wesley Ivan Hurt, Nov 15 2014
  • Mathematica
    For[n=1,n<31,n++,Print[2^(2*n)-(-1)^(n)*Binomial[2n,n]]]

Formula

a(n) = 2^(2n) - (-1)^n * binomial(2n, n).
n*(4*n-5)*a(n) +2*(-4*n+3)*a(n-1) -8*(4*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jun 09 2018

A247984 Constant terms of congruences derived from a permutation polynomial motivated by generalized quadrangles.

Original entry on oeis.org

2, 6, 8, 10, 32, 84, 128, 186, 512, 1276, 2048, 3172, 8192, 19816, 32768, 52666, 131072, 310764, 524288, 863820, 2097152, 4899736, 8388608, 14073060, 33554432, 77509464, 134217728, 228318856, 536870912, 1228859344, 2147483648
Offset: 1

Author

Brian G. Kronenthal, Sep 28 2014

Keywords

Comments

Let p be an odd prime, let e be a positive integer, and let q = p^e. Dmytrenko, Lazebnik, and Williford (2007) proved that every monomial graph of girth at least eight is isomorphic to G = G_q(xy, x^ky^(2k)) for some integer k which is not divisible by p. If q = 3, then G is isomorphic to G_3(xy, xy^2). If q >= 5, then F(x) = ((x + 1)^(2k) - 1)x^(q - 1 - k) - 2x^(q - 1) is a permutation polynomial, in which case the Hermite-Dickson Criterion implies that the coefficient at x^(q - 1) in F(x)^n must equal 0 modulo p. a(n) is the constant term of the coefficient at x^(q - 1) in F(x)^n; this was first stated in Kronenthal (2012). The provided Mathematica program produces the first 30 terms of the sequence.

Crossrefs

Cf. A246800.

Programs

  • Maple
    A247984 := proc(n)
        if type(n,'odd') then
            2^n;
        else
            2^n-(-1)^(n/2)*binomial(n, n/2) ;
        end if;
    end proc: # R. J. Mathar, Jun 09 2018
  • Mathematica
    For[n = 1, n < 31, n++, Piecewise[{{Print[2^n - (-1)^(n/2) * Binomial[n, n/2]], EvenQ[n]}, {Print[2^n], OddQ[n]}}]]
    Rest[With[{nn = 50}, CoefficientList[Series[Exp[2*x] - BesselJ[0, 2*x], {x, 0, nn}], x]*Range[0, nn]!]] (* G. C. Greubel, Aug 16 2017 *)
  • PARI
    a(n) = if (n % 2, 2^n, 2^n - (-1)^(n/2)*binomial(n, n/2)); \\ Michel Marcus, Oct 01 2014

Formula

a(n) = 2^n when n is odd and a(n) = 2^n - (-1)^(n/2)*binomial(n, n/2) when n is even.
From Robert Israel, Oct 01 2014: (Start)
G.f.: 1/(1-2*x) - 1/sqrt(1+4*x^2).
E.g.f.: exp(2*x) - J_0(2*x) where J_0 is a Bessel function. (End)
n*(2*n-3)*a(n) -2*(2*n-1)*(n-1)*a(n-1) +4*(n-1)*(2*n-3)*a(n-2) -8*(2*n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 09 2018