cp's OEIS Frontend

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

%I A246800 #30 Sep 08 2022 08:46:09
%S A246800 6,10,84,186,1276,3172,19816,52666,310764,863820,4899736,14073060,
%T A246800 77509464,228318856,1228859344,3693886906,19513475404,59644341436,
%U A246800 310223170744,961665098956,4936304385544,15488087080696,78602174905264,249227373027556,1252310513280376,4007681094422392,19961337935130096,64408903437167496,318297642651252784,1034656923041985552
%N A246800 Even-indexed terms of A247984, a sequence motivated by generalized quadrangles.
%C A246800 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.
%H A246800 V. Dmytrenko, F. Lazebnik, and J. Williford, <a href="http://dx.doi.org/10.1016/j.ffa.2006.03.001">On monomial graphs of girth eight</a>, Finite Fields and Their Applications 13 (2007), 828-842.
%H A246800 Brian G. Kronenthal, <a href="http://dx.doi.org/10.1016/j.ffa.2012.01.001">Monomial Graphs and Generalized Quadrangles</a>, Finite Fields and Their Applications, 18 (2012), 674-684.
%H A246800 B. G. Kronenthal, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kronenthal/kron3.html">An Integer Sequence Motivated by Generalized Quadrangles</a>, Journal of Integer Sequences, 2015, Vol. 18. #15.7.8.
%F A246800 a(n) = 2^(2n) - (-1)^n * binomial(2n, n).
%F A246800 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
%p A246800 A246800:=n->2^(2*n)-(-1)^n*binomial(2*n, n): seq(A246800(n), n=1..30); # _Wesley Ivan Hurt_, Nov 15 2014
%t A246800 For[n=1,n<31,n++,Print[2^(2*n)-(-1)^(n)*Binomial[2n,n]]]
%o A246800 (Magma) [2^(2*n)-(-1)^n*Binomial(2*n, n) : n in [1..30]]; // _Wesley Ivan Hurt_, Nov 15 2014
%Y A246800 Equals even-indexed terms of A247984.
%K A246800 easy,nonn
%O A246800 1,1
%A A246800 _Brian G. Kronenthal_, Nov 15 2014