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

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

• 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. 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

• 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

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