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, 3264, 168399, 2584576, 21328125, 119952576, 518949739, 1853620224, 5718836601, 15715000000, 39312710151, 90985918464, 197228242549, 404268317376, 789541171875, 1478257278976, 2666742760689, 4654606866624, 7888229913151, 13018560000000, 20976763640301, 33071897577664, 51115460148699

Offset

1,2

References

William Fulton, Intersection Theory 2nd edition, Springer-Verlag, 1998, page 192.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

David Eisenbud and Brady Haran, The Journey to 3264, Numberphile video (2023).

Wikipedia, Steiner's conic problem.

Wikipedia, Enumerative geometry.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

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

From Elmo R. Oliveira, Jun 02 2026: (Start)

G.f.: x*(1 + 3253*x + 132550*x^2 + 911542*x^3 + 1621504*x^4 + 785704*x^5 + 127402*x^6 + 41530*x^7 + 5263*x^8 + 51*x^9) / (1 - x)^11.

E.g.f.: exp(x)*x*(1 + 1631*x + 26435*x^2 + 80440*x^3 + 83805*x^4 + 38102*x^5 + 8440*x^6 + 940*x^7 + 50*x^8 + x^9).

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). (End)

Example

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

Mathematica

A328148[n_] := n^6*(n*(n*(n*(n + 5) + 10) - 30) + 15);
Array[A328148, 25] (* Paolo Xausa, Jul 06 2026 *)

Crossrefs

Sequence in context: A358870 A337792 A251277 * A286008 A220594 A101706

Adjacent sequences: A328145 A328146 A328147 * A328149 A328150 A328151

Keyword

nonn,easy,changed

Author

Niccolò Castronuovo, Jun 07 2020

Extensions

More terms from Paolo Xausa, Jul 06 2026

Status

approved