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A049114 2-ranks of difference sets constructed from Glynn type II hyperovals.

%I A049114 #14 Sep 08 2022 08:44:58
%S A049114 1,1,5,7,21,37,89,173,383,777,1665,3441,7277,15159,31885,66645,139865,
%T A049114 292757,613823,1285585,2694433,5644609,11828501,24782311,51928773,
%U A049114 108802597,227978105,477674813,1000877759,2097121497,4394101857
%N A049114 2-ranks of difference sets constructed from Glynn type II hyperovals.
%H A049114 G. C. Greubel, <a href="/A049114/b049114.txt">Table of n, a(n) for n = 1..1000</a>
%H A049114 R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, <a href="http://dx.doi.org/10.1006/jcta.1998.2950">Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets</a>, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
%H A049114 Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, <a href="https://www.mat.univie.ac.at/~kratt/artikel/glynn.html">Supplement to "Gauss Sums, Jacobi Sums and p-ranks ..."</a>
%H A049114 Q. Xiang, <a href="https://doi.org/10.1109/18.737547">On Balanced Binary Sequences with Two-Level Autocorrelation Functions</a>, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.
%F A049114 G.f.: (1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5).
%F A049114 a(n+1) = a(n) + 3*a(n-1) - a(n-2) - a(n-3) + 1.
%p A049114 L := 1,1,5,7: for i from 5 to 100 do l := nops([ L ]): L := L,op(l,[ L ])+3*op(l-1,[ L ])-op(l-2,[ L ])-op(l-3,[ L ])+1: od: [ L ];
%t A049114 Join[{1,1,5,7}, Table[a[1]=1; a[2]=1; a[3]=5; a[4]=7; a[i]=a[i-1]+ 3*a[i-2]-a[i-3]-a[i-4] +1, {i, 5, 40}]]
%t A049114 CoefficientList[Series[(1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5), {x, 0, 40}], x] (* _G. C. Greubel_, Jul 10 2019 *)
%o A049114 (PARI) my(x='x+O('x^40)); Vec((1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5)) \\ _G. C. Greubel_, Jul 10 2019
%o A049114 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5) )); // _G. C. Greubel_, Jul 10 2019
%o A049114 (Sage) ((1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 10 2019
%o A049114 (GAP) a:=[1,5,7,21];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-a[n-3] -a[n-4] +1; od; Concatenation([1], a); # _G. C. Greubel_, Jul 10 2019
%Y A049114 Cf. A001595, A049112.
%K A049114 nonn,easy
%O A049114 1,3
%A A049114 Christian Krattenthaler (kratt(AT)ap.univie.ac.at)