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A246800 Even-indexed terms of A247984, a sequence motivated by generalized quadrangles. 2
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 (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. 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.

LINKS

Table of n, a(n) for n=1..30.

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^(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

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]]]

PROG

(MAGMA) [2^(2*n)-(-1)^n*Binomial(2*n, n) : n in [1..30]]; // Wesley Ivan Hurt, Nov 15 2014

CROSSREFS

Equals even-indexed terms of A247984.

Sequence in context: A166386 A280881 A256246 * A269341 A201921 A117310

Adjacent sequences:  A246797 A246798 A246799 * A246801 A246802 A246803

KEYWORD

easy,nonn

AUTHOR

Brian G. Kronenthal, Nov 15 2014

STATUS

approved

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Last modified June 15 04:47 EDT 2021. Contains 345043 sequences. (Running on oeis4.)