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

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

• easy
• nonn

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• V. Dmytrenko, F. Lazebnik, and J. Williford, On monomial graphs of girth eight, Finite Fields and Their Applications 13 (2007), 828-842.
• Brian G. Kronenthal, Monomial Graphs and Generalized Quadrangles, Finite Fields and Their Applications, 18 (2012), 674-684.
• B. G. Kronenthal, An Integer Sequence Motivated by Generalized Quadrangles, Journal of Integer Sequences, 2015, Vol. 18. #15.7.8.

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