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A019590 Fermat's Last Theorem: a(n) = 1 if x^n + y^n = z^n has a nontrivial solution in integers, otherwise a(n) = 0. 119
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) is the Hankel transform of A000045(n), n>=1 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - Wolfdieter Lang, Jan 23 2007
1, -1, 0, 0, 0, ... is the convolutional inverse of the all-ones sequence. - Tanya Khovanova, Jun 29 2007
Also parity of the Euler totient function A000010. - Omar E. Pol, Jan 15 2012
a(n-1) gives the row sums of A048994. - Wolfdieter Lang, May 09 2017
Decimal expansion of 11/10. - Franklin T. Adams-Watters, Mar 08 2019
REFERENCES
A. D. Aczel, Fermat's Last Theorem, Four Walls Eight Windows NY 1996.
A. C. Clarke, The Last Theorem, Gollancz SF 2004.
B. Cipra, What's Happening in the Mathematical Sciences 1994 Vol. 2, "A Truly Remarkable Proof" pp. 3-8 AMS Providence RI.
B. Cipra, What's Happening in the Mathematical Sciences 1995-6 Vol. 3, 'Fermat's Theorem-At Last' pp. 2-13 AMS Providence RI.
J. Coates and S.-T. Yau (Eds), Elliptic Curves, Modular Forms and Fermat's last Theorem, International Press Boston MA 1998.
G. Cornell, J. H. Silverman and G. Stevens (Eds), Modular Forms and Fermat's last Theorem, Springer NY 2000.
K. J. Devlin, Mathematics: The New Golden Age, Chapter 10, Columbia Univ. Press NY 1999.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 731.
H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
G. Giorello & C. Sinigaglia, "Fermat: De défis en conjectures", Les génies de la science No. 32 Aug-Oct 2007, pp. 82-100, Pour la Science, Paris.
C. Goldstein, "Le Théorème de Fermat", La Recherche, Vol. Mar 25 1994, pp. 268-275, Paris.
C. Goldstein, "Le Théorème de Fermat enfin démontré", Chapter IX pp. 111-129 in 'Histoire Des Nombres', La Recherche, Tallandier, Paris 2007.
Y. Hellegouarch, "Fermat Vaincu", Quadrature No. 22 pp. 37-55 Editions du choix Argenteuil (France) 1995.
Y. Hellegouarch, "Fermat enfin démontré", Pour la Science, No. 220, 1996 pp. 92-97 Paris.
Y. Hellegouarch, Invitation aux mathématiques de Fermat-Wiles, Dunod Paris 2001.
Y. Hellegouarch, "L'Enigme du Theoreme de Fermat" pp. 31-41 in 'Qu'est-ce que l'Univers?", Université de tous les savoirs, Vol. 4 (Edit. Y. Michaud), Odile Jacob Paris 2001.
Y. Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic Press NY 2001.
P. Hoffman, The Man Who Loved Only Numbers, pp. 183-200, Hyperion NY 1998
W. Lindsay, Fermat's Last Theorem, A Perfect Proof, Lulu Press, Morrisville NC 2005.
L. J. Mordell, Three lectures on Fermat's last theorem, Cambr. Univ. Press 1921 (Reprinted by The Scholarly Pub. Office, Univ. of Michigan Library 2005).
C. J. Mozzochi, The Fermat Diary, AMS Providence RI 2000.
C. J. Mozzochi, The Fermat Proof, New Bern NC 2004.
V. K. Murty, Seminar on Fermat's Last Theorem, Amer. Math. Soc. Providence RI 1995.
P. Odifreddi, The Mathematical Century, Chapter 2.14 "Number Theory: Wiles' Proof of Fermat's Last Theorem (1995)" p. 82 Princeton Univ. Press NJ 2004.
I. Peterson, The Mathematical Tourist, pp. 234-238, W. H. Freeman/Owl Book NY 2001.
I. Peterson, "A Marginal Note" in Islands of Truths, pp. 280-285, W. H. Freeman NY 1990.
A. van der Poorten, Notes on Fermat's Last Theorem, Wiley NY 1996
J. Propp, Who Proved Fermat's Last Theorem? Princeton Univ. Press NJ 2005.
P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979.
P. Ribenboim, Fermat's Last Theorem for Amateurs, Springer Verlag NY 1999.
R. Schoof, "Wiles' proof of the Taniyama-Weil conjecture for semi-stable elliptic curves over Q", Chap. 14 in 'Ou En Sont Les Mathématiques ?' Soc. Math. de France (SMF), Vuibert, Paris 2002.
S. Singh, Fermat's Enigma, Walker and Co. NY 1997.
I. Stewart, From Here To Infinity, Chapter 3 "Marginal Interest" pp. 25-48 OUP Oxford 1996.
I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem, A. K. Peters Natick MA 2001.
G. R. Talbott, Fermat's Last Theorem, Lotus Press WI 1991.
R. Van Vo, Fermat's Last Theorem, AuthorHouse, Bloomington IN 2002.
J. Vigouroux et al., Une aventure mathématique, le théorème de Fermat, BT2 series No. 6, PEMF Mouans-Sartoux(France) 1998.
LINKS
Amer. Math. Soc., Fermat's Last Theorem
Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
A. J. Bailey, Fermat's Last Theorem
K. Belabas and C. Goldstein, Fermat et son Theoreme
C. K. Caldwell, The Prime Glossary, Fermat's last theorem
A. Camus College Team, La conjecture de Fermat
Larry Freeman's Blog Spot, Fermat's Last Theorem
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
S. Horsler, Home Page on FLT
A. Krowne, PlanetMath.org, Fermat's last theorem
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
John Lynch (writer/producer) and Simon Singh (director), Fermat's Last Theorem. Horizon. BBC TV/WGBH Boston, Jan 15 1996.
M. Marcionelli, La conjecture de Fermat
Mathematical Database, Poster, Fermat's Last Theorem
J. S. Milne, Elliptic Curves
J. J. O'Connor and E. F. Robertson, Fermat's Last Theorem
K. Rubin and A. Silverberg, A Report on Wiles' Cambridge Lectures
K. Rubin and A. Silverberg, A report on Wiles' Cambridge lectures, arXiv:math/9407220 [math.NT], 1994.
K. Rubin and A. Silverberg, A Report On Wiles' Cambridge Lectures, Bull. Amer. Math. Soc. 31 (1994), 15-38.
D. Rusin, The Mathematical Atlas, Higher degree equations; Fermat's equation [Broken link]
D. Rusin, The Mathematical Atlas, Higher degree equations; Fermat's equation [Cached copy, but just of the top page, so none of the internal links will work]
School of Mathematics and Statistics, University of St Andrews, Fermat's last theorem.
J. L. Selfridge, C. A. Nicol and H. S. Vandiver, Proof Of Fermat's Last Theorem For All Prime Exponents Less Than 4002
S. Singh, Fermat Corner
Ian Stewart, Fermat's Last Time-Trip, in Scientific American Nov. 1993 pp. 85.
D. Surendran, Fermat's Last Theorem
C. Thornhill, Fermat's Last Theorem
A. van der Poorten, Fermat's Last Theorem
G. Villemin's Almanach of Numbers, Theoreme de Fermat-Wiles
A. J. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), 443-551, doi:10.2307/2118559.
Yan X Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.
FORMULA
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} (-1)^A010060(n-2k) mod (C(n, 2k), 2). - Paul Barry, Jan 03 2005
Euler transform of length 2 sequence [1, -1]. - Michael Somos, Jul 05 2009
a(n) is multiplicative with a(2) = 1, a(2^e) = 0 if e > 1, a(p^e) = 0^e if p > 2. - Michael Somos, Jul 05 2009
G.f.: x + x^2 = x * (1 - x^2) / (1 - x). - Michael Somos, Jul 05 2009
Dirichlet g.f.: 1 + 2^(-s). - Michael Somos, Jul 05 2009
a(n) = A000035(A000010(n)). - Omar E. Pol, Oct 28 2013
PROG
(PARI) {a(n) = (n==1) + (n==2)}; /* Michael Somos, Jul 05 2009 */
CROSSREFS
INVERT transform gives Fibonacci numbers, A000045.
Convolution inverse of A062157. Dirichlet convolution inverse of A154269.
Cf. A229382, A229383 (near-miss counterexamples to FLT).
Cf. A048994 (row sums).
Sequence in context: A134323 A060576 A261012 * A154955 A240356 A240354
KEYWORD
nonn,nice,easy,mult
AUTHOR
STATUS
approved

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