A000905 Hamilton numbers.

2, 3, 5, 11, 47, 923, 409619, 83763206255, 3508125906290858798171, 6153473687096578758448522809275077520433167, 18932619208894981833333582059033329370801266249535902023330546944758507753065602135843

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

a(n) is the minimal degree of an equation from which n successive terms after the first can be removed (by a series of transformation comparable to Tschirnhaus') without requiring the solution of an equation of degree greater than n (and excluding cases where an equation of degree greater than n is needed but is in fact factorizable into several equations of degree all less than n). Hamilton computed the first six terms of this sequence (see reference). That is the reason Sylvester and Hammond named them "Hamilton numbers". - Olivier Gérard, Oct 17 2007

Named after the Irish mathematician William Rowan Hamilton (1805-1865). - Amiram Eldar, Jun 19 2021

Examples

			a(1)=2 is the familiar fact than one can always remove the linear term of a quadratic equation.
a(2)=3 because one can put any cubic equation in the form x^3-a=0 by a Tschirnhaus transformation based on the solutions of a quadratic equation.
a(4)=11 because one can remove the 4 terms after the first term in a polynomial of degree 11 without having to solve a quintic.
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..14
• Tigran Ananyan and Melvin Hochster, Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers, arXiv:2003.00589 [math.AC], 2020.
• Alexander Chen, Y. H. He and J. McKay, Erland Samuel Bring's "Transformation of Algebraic Equations", arXiv preprint arXiv:1711.09253 [math.HO], 2017. See page 6.
• Raymond Garver, The Tschirnhaus transformation, The Annals of Mathematics, 2nd Ser., Vol. 29, No. 1/4. (1927 - 1928), pp. 329.
• William Rowan Hamilton, Inquiry Into the Validity of a Method Recently Proposed by George B. Jerrard, Esq. for Transforming and Resolving Equations of Elevated Degrees Undertaken at the Request of the Association. Richard and John E. Taylor, Report of the Sixth Meeting of the British Association for the Advancement of Science, held at Bristol in August 1836 (London: John Murray, Albemarle St., 1837), pp. 295-348.
• Edouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 496.
• Edouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1. [Annotated scan of pages 488-499 only]
• Alexander J. Sutherland, Upper Bounds on Resolvent Degree and Its Growth Rate, arXiv:2107.08139 [math.AG], 2021.
• James Joseph Sylvester and James Hammond, On Hamilton's numbers, Philosophical Transactions of the Royal Society of London A, Vol. 178 (1887), pp. 285-312, alternative link.
• Jesse Wolfson, Tschirnhaus transformations after Hilbert, arXiv:2001.06515 [math.AG], 2020.

Crossrefs

Cf. A001660.

Equals A006719(n) - 1.

Cf. A134294.

Programs

Maple

A000905 := proc(n) option remember; local i; if n=1 then 2 else 2+add((-1)^(i+1)*binomial(A000905(n-i),i+1),i=1..n-1); fi; end;

Mathematica

a[1]=2; a[n_] := a[n] = 2+Sum[(-1)^(i+1)*Product[a[n-i] - k, {k, 0, i}]/(i+1)!, {i, 1, n-1}]; Table[a[n], {n, 1, 11}] (* Jean-François Alcover, May 17 2011, after Maple prog. *)

Python

from sympy import binomial
a = [2]
[a.append(2+sum((-1)**(i)*binomial(a[n-i-1], i+2) for i in range(n))) for n in range(1,11)]
print(a) # Nicholas Stefan Georgescu, Mar 01 2023

Extensions

The formula given by Lucas on p. 498 is slightly in error - see Maple program given here.