Adam M. Scherlis - cp's OEIS frontend

User: Adam M. Scherlis

Adam M. Scherlis has authored 3 sequences.

A376922 Variance of n-th power of a standard normal random variable.

1, 2, 15, 96, 945, 10170, 135135, 2016000, 34459425, 653836050, 13749310575, 316126087200, 7905853580625, 213439785208650, 6190283353629375, 191894675132160000, 6332659870762850625, 221641908024728441250, 8200794532637891559375, 319830558102716120460000

Offset: 1

Adam M. Scherlis, Oct 10 2024

nonn

The variance of the n-th sample moment is exactly a(n) / k for sample size k.

For a non-standard normal r.v. X ~ N(0, sigma^2), Var(X^n) = a(n) sigma^(2n).

			If Z ~ N(0, 1), then Var(Z) = 1, Var(Z^2) = 2, Var(Z^3) = 15, etc.

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics Volume 1, Charles Griffin & Company, 1963, page 229.

Cf. A001147, A103639, A123023.

a(n) = M(2n) - M(n)^2, where M(n) = A123023(n) are the moments of N(0, 1).

A370459 Number of unicursal stars with n vertices.

Original entry on oeis.org

0, 0, 1, 1, 5, 19, 112, 828, 7441, 76579, 871225, 10809051, 144730446, 2079635889, 31912025537, 520913578812, 9013780062785, 164829273635749, 3176388519597555, 64343477504391475, 1366925655386979893, 30390554390984325019, 705740995420852895453

Offset: 3

Adam M. Scherlis, Feb 19 2024

nonn

A unicursal star is a closed loop formed by diagonals of a regular n-gon.
These are Hamiltonian cycles on the graph complement of the n-cycle.
Allowing polygon diagonals, but not sides, is equivalent to requiring every edge to cross at least one other edge.
These are counted up to rotation and reflection, i.e., modulo dihedral symmetry of the n-gon.
Inspired by a unicursal dodecagram drawn by Gordon FitzGerald (see links).

			For n=5, there is only the regular pentagram {5/2}.
For n=6, there is only the unicursal hexagram.
For n=7, in addition to the two regular heptagrams {7/2} and {7/3}, there are three nontrivial unicursal heptagrams represented by:
 (0, 2, 4, 1, 6, 3, 5, 0)
 (0, 2, 5, 1, 3, 6, 4, 0)
 (0, 2, 5, 1, 4, 6, 3, 0).

Andrew Howroyd, Table of n, a(n) for n = 3..200
Gordon FitzGerald, illustration of a unicursal dodecagram.
Wikipedia, Unicursal hexagram.

Cf. A000940 (polygon sides allowed).
Cf. A055684 (cases with dihedral symmetry only).
Cf. A002816 (rotations and reflections counted separately).
Cf. A231091 (up to rotations only), A370769 (achiral).

\\ Requires a370068 from A370068.
Ro(n)=-(-1)^n + subst(serlaplace(polcoef(((1 - x)^2)/(2*(1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
Re(n)=subst(serlaplace(polcoef((1 - x - 2*x^2)/(4*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
a(n)={if(n<3, 0, (if(n%2, 2*Ro(n\2), Re(n/2)) + a370068(n))/4)} \\ Andrew Howroyd, Mar 01 2024

a(n) = (A231091(n) + A370769(n))/2. - Andrew Howroyd, Mar 06 2024

a(14) onwards from Andrew Howroyd, Feb 26 2024

A309541 Representable positive integers n that are not the inverse of their inverse in binary64 (double precision) IEEE 754 floating-point arithmetic (Version where 1 and n*(1/n) are unequal).

Original entry on oeis.org

49, 98, 103, 107, 161, 187, 196, 197, 206, 214, 237, 239, 249, 253, 322, 347, 374, 389, 392, 394, 412, 417, 425, 428, 443, 474, 478, 479, 491, 498, 499, 501, 503, 506, 509, 561, 569, 644, 685, 691, 694, 725, 729, 735, 737, 748, 753, 765, 778, 779, 784, 788, 789, 797, 809, 817, 823, 824, 829, 833, 834, 837, 841, 849, 850

Offset: 1

Adam M. Scherlis, Aug 06 2019

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Efstratios Gallopoulos, Scientific Computation I, five terms provided (in Greek) [broken link]
Mark, comment on blog post "Rundungsfehler im Flash Player?", eight terms (in German)
koDoz, comment in thread "Welche Grenzwerte gelten für Zahlen, Strings usw?", nine terms (in German)

See A275419 for the n != 1/(1/n) version.

doubleprecision one, r
      integer i
      parameter (one=1.0D0)
      do 10 i = 1, 500
      R = one / dble(i)
      if ( R * dble(i) .ne. one) write (*,1000) i
1000  format (i0)
10    continue
      end
C Hugo Pfoertner, Jan 18 2024

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User: Adam M. Scherlis

A376922 Variance of n-th power of a standard normal random variable.

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A370459 Number of unicursal stars with n vertices.

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A309541 Representable positive integers n that are not the inverse of their inverse in binary64 (double precision) IEEE 754 floating-point arithmetic (Version where 1 and n*(1/n) are unequal).

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