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A338551 Number of ways to make a checkout score of n in darts.

0, 1, 1, 4, 7, 14, 20, 31, 39, 55, 65, 86, 96, 126, 133, 171, 179, 223, 228, 286, 283, 352, 348, 422, 408, 497, 467, 569, 534, 642, 594, 720, 654, 791, 719, 863, 775, 942, 831, 1012, 894, 1082, 945, 1159, 991, 1216, 1037, 1263, 1062, 1311, 1081, 1340, 1110, 1366

Offset: 1

Carmen Bruni, Nov 02 2020

In other words, the number of ways to achieve a score of n using at most 3 darts and finishing on a double. The maximum checkout score is 170, so this is a finite sequence.

Carmen Bruni, Table of n, a(n) for n = 1..170
Wikipedia, Darts

Cf. A008575, A242718, A242678, A242681, A167213, A076119, A244512, A117883.

seq()={my(s=x*(1-x^20)/(1-x)+x^25, d=subst(s,x,x^2), g=s+d+subst(s-x^25,x,x^3)); Vecrev((1+g+g^2)*d/x)} \\ Andrew Howroyd, Nov 04 2020

def darts(n):
  if n > 170 or n <= 1:
    return 0
  ans = 0
  singles = list(range(1, 21)) + [25]
  doubles = list(map(lambda x: 2*x, singles))
  triples = list(map(lambda x: 3*x, singles[:-1]))
  throws = singles+doubles+triples
  for i in range(len(throws)):
    for j in range(len(throws)):
      for k in range(len(doubles)):
        dart1 = throws[i]
        dart2 = throws[j]
        dart3 = doubles[k]
        if dart1 + dart2 + dart3 == n:
          ans += 1
    for j in range(len(doubles)):
      dart1 = throws[i]
      dart2 = doubles[j]
      if dart1 + dart2 == n:
        ans += 1
  return ans + (n in doubles)
for i in range(1,171):
  print(darts(i))

A212420 Known primes such that there are no pairwise coprime solutions to the Diophantine equation of the form x^3 + y^3 = p^a z^n with a >= 1 an integer and n >= p^(2p) prime.

Original entry on oeis.org

53, 83, 149, 167, 173, 199, 223, 227, 233, 263, 281, 293, 311, 347, 353, 359, 389, 401, 419, 443, 449, 461, 467, 479, 487, 491, 563, 569, 571, 587, 599, 617, 641, 643, 659, 719, 727, 739, 743, 751, 809, 811, 823, 827, 829, 839, 859, 881, 887, 907, 911, 929, 941, 947, 953, 977, 983

Offset: 1

Carmen Bruni, May 15 2012

These primes are the prime numbers p greater than 3 such that for every elliptic curves with conductor of the form 18p, 36p, or 72p we have that 4 does not divide the order of the torsion subgroup over the rationals but at least one curve with 2 dividing this order, such that there is a prime q congruent to 1 modulo 6 such that 4 does not divide the order of the torsion subgroup over the finite field of size q.

M. A. Bennett, F. Luca and J. Mulholland, Twisted extensions of the cubic case of Fermat's Last Theorem, Ann. Sci. Math. Quebec. 35 (2011), 1-15.

A196995 Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1.

Original entry on oeis.org

0, -128, -5038848, 140737488355328, 5000000000000000000000000, -354400937492545922690672153504784580608, -72317557999158469111384459491956546088110808312359944192, 57896044618658097711785492504343953926634992332820282019728792003956564819968

Offset: 1

Carmen Bruni, Oct 08 2011

K(x,y) = 2n*Tr(xy)

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, 21-22

interface(rtablesize=infinity):
with(LinearAlgebra):
for n from 1 to 12 do
for i from 1 by 1 to n-1 do
   M[i] := Matrix(n);
   M[i](i,i) := 1;
   M[i](i+1,i+1) := -1;
  end do:
  ctr := n:
  for i from 1 by 1 to n do
  for j from 1 by 1 to n do
  if(i <> j) then
    M[ctr] := Matrix(n);
    M[ctr](i,j) := 1;
    ctr := ctr +1;
  end if
  end do:
end do:
A := Matrix(n^2-1):
for i from 1 by 1 to n^2-1 do
  for j from 1 by 1 to n^2-1 do
   A(i,j) := 2*n*Trace(M[i].M[j]):
  end do:
  end do:
  print(Determinant(A));
end do:
# Alternatively, using the second description
  print(0);
  for n from 2 to 20 do
  print((-1)^(binomial(n,2))*2^(n^2-1)*n^(n^2));
  end do:

a(n) = (-1)^binomial(n,2) *2^(n^2-1)*n^(n^2) for n>= 2

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User: Carmen Bruni

A338551 Number of ways to make a checkout score of n in darts.

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A212420 Known primes such that there are no pairwise coprime solutions to the Diophantine equation of the form x^3 + y^3 = p^a z^n with a >= 1 an integer and n >= p^(2p) prime.

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A196995 Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1.

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