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A247984 Constant terms of congruences derived from a permutation polynomial motivated by generalized quadrangles. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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

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 *)

PROG

(PARI) a(n) = if (n % 2, 2^n, 2^n - (-1)^(n/2)*binomial(n, n/2)); \\ Michel Marcus, Oct 01 2014

CROSSREFS

Cf. A246800.

Sequence in context: A066762 A066198 A299381 * A108417 A185078 A214402

Adjacent sequences:  A247981 A247982 A247983 * A247985 A247986 A247987

KEYWORD

easy,nonn

AUTHOR

Brian G. Kronenthal, Sep 28 2014

STATUS

approved

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Last modified July 19 09:50 EDT 2018. Contains 312774 sequences. (Running on oeis4.)