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A328148 The number of conics tangent to five non-singular curves of degree n in general position, in a projective plane defined over an algebraically closed field of characteristic zero. 1

%I #51 Apr 04 2023 22:19:21

%S 1,3264,168399,2584576,21328125,119952576,518949739,1853620224,

%T 5718836601,15715000000,39312710151,90985918464,197228242549,

%U 404268317376,789541171875,1478257278976,2666742760689,4654606866624,7888229913151,13018560000000

%N The number of conics tangent to five non-singular curves of degree n in general position, in a projective plane defined over an algebraically closed field of characteristic zero.

%D W. Fulton, Intersection Theory 2.ed., Springer-Verlag, 1998, page 192.

%H David Eisenbud and Brady Haran, <a href="https://www.youtube.com/watch?v=NWahomDHaDs">The Journey to 3264</a>, Numberphile video (2023).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Steiner%27s_conic_problem">Steiner's conic problem</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Enumerative_geometry">Enumerative Geometry</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F a(n) = n^5 * ((n-1)^5 + 10*(n-1)^4 + 40*(n-1)^3 + 40*(n-1)^2 + 10*(n-1) + 1).

%e a(2)=3264 because there are 3264 conics tangent to five conics in general position (Steiner's conic problem).

%K nonn,easy

%O 1,2

%A _Niccolò Castronuovo_, Jun 07 2020

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