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A000905 Hamilton numbers.

%I A000905 M0736 N0275 #77 Mar 01 2023 15:33:46
%S A000905 2,3,5,11,47,923,409619,83763206255,3508125906290858798171,
%T A000905 6153473687096578758448522809275077520433167,
%U A000905 18932619208894981833333582059033329370801266249535902023330546944758507753065602135843
%N A000905 Hamilton numbers.
%C A000905 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
%C A000905 Named after the Irish mathematician William Rowan Hamilton (1805-1865). - _Amiram Eldar_, Jun 19 2021
%D A000905 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000905 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000905 Amiram Eldar, <a href="/A000905/b000905.txt">Table of n, a(n) for n = 1..14</a>
%H A000905 Tigran Ananyan and Melvin Hochster, <a href="https://arxiv.org/abs/2003.00589">Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers</a>, arXiv:2003.00589 [math.AC], 2020.
%H A000905 Alexander Chen, Y. H. He and J. McKay, <a href="https://arxiv.org/abs/1711.09253">Erland Samuel Bring's "Transformation of Algebraic Equations"</a>, arXiv preprint arXiv:1711.09253 [math.HO], 2017. See page 6.
%H A000905 Raymond Garver, <a href="http://www.jstor.org/stable/1968002">The Tschirnhaus transformation</a>, The Annals of Mathematics, 2nd Ser., Vol. 29, No. 1/4. (1927 - 1928), pp. 329.
%H A000905 William Rowan Hamilton, <a href="https://emis.de/classics/Hamilton/Jerrard.pdf">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</a>, 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.
%H A000905 Edouard Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 496.
%H A000905 Edouard Lucas, <a href="/A000904/a000904.pdf">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1. [Annotated scan of pages 488-499 only]
%H A000905 Alexander J. Sutherland, <a href="https://arxiv.org/abs/2107.08139">Upper Bounds on Resolvent Degree and Its Growth Rate</a>, arXiv:2107.08139 [math.AG], 2021.
%H A000905 James Joseph Sylvester and James Hammond, <a href="http://www.jstor.org/stable/90558">On Hamilton's numbers</a>, Philosophical Transactions of the Royal Society of London A, Vol. 178 (1887), pp. 285-312, <a href="https://archive.org/details/philtrans02248142">alternative link</a>.
%H A000905 Jesse Wolfson, <a href="https://arxiv.org/abs/2001.06515">Tschirnhaus transformations after Hilbert</a>, arXiv:2001.06515 [math.AG], 2020.
%e A000905 a(1)=2 is the familiar fact than one can always remove the linear term of a quadratic equation.
%e A000905 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.
%e A000905 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.
%p A000905 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;
%t A000905 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. *)
%o A000905 (Python)
%o A000905 from sympy import binomial
%o A000905 a = [2]
%o A000905 [a.append(2+sum((-1)**(i)*binomial(a[n-i-1], i+2) for i in range(n))) for n in range(1,11)]
%o A000905 print(a) # _Nicholas Stefan Georgescu_, Mar 01 2023
%Y A000905 Cf. A001660.
%Y A000905 Equals A006719(n) - 1.
%Y A000905 Cf. A134294.
%K A000905 nonn,nice,easy
%O A000905 1,1
%A A000905 _N. J. A. Sloane_
%E A000905 The formula given by Lucas on p. 498 is slightly in error - see Maple program given here.