Search: id:A001078
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%I A001078 M2122 N0839
%S A001078 0,2,20,198,1960,19402,192060,1901198,18819920,186298002,1844160100,
%T A001078 18255302998,180708869880,1788833395802,17707625088140,175287417485598,
%U A001078 1735166549767840,17176378080192802,170028614252160180,1683109764441408998
%N A001078 a(n) = 10*a(n-1)-a(n-2) with a(0) = 0, a(1) = 2.
%C A001078 Also 6*x^2+1 is a square. - Cino Hilliard (hillcino368(AT)gmail.com), 
               Mar 08 2003
%C A001078 This sequence has the following property. For each n, if A = a(n), B 
               = 2*a(n+1), C = 3*a(n+1) then A*B+1, A*C+1, B*C+1 are perfect squares. 
               - Deshpande M.N. (dpratap_ngp(AT)sancharnet.in), Sep 22 2004
%C A001078 n such that 6*n^2=floor(sqrt(6)*n*ceil(sqrt(6)*n)). - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), May 10 2003
%C A001078 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jun 19 2005
%C A001078 (sqrt(2)+sqrt(3))^(2*n)=A001079(n)+a(n)*sqrt(6); a(n)=A054320(n)+A138288(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008
%D A001078 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001078 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001078 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A001078 O. Bottema: Verscheidenheden XXVI. Het vraagstuk van Malfatti, Euclides 
               25 (1949-50), pp. 144-149 [in Dutch].
%D A001078 O. Bottema, The Malfatti problem (translation of Het vraagstuk van Malfatti), 
               Forum Geom. 1 (2001) 43-50.
%D A001078 V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, 
               Paris, 1952, p. 281.
%D A001078 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 
               283, 302, P_{16}).
%H A001078 T. D. Noe, <a href="b001078.txt">Table of n, a(n) for n=0..100</a>
%H A001078 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001078 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001078 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001078 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001078 <a href="http://forumgeom.fau.edu/FG2001volume1/FG200107index.html">Bottema 
               article in Forum Geometricorum</a>
%H A001078 <a href="malfatti.html">Bottema article, from Euclides</a>
%H A001078 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">
               De solutione problematum diophanteorum per numeros integros</a>, 
               par. 18
%F A001078 a(n)=[(5+2sqrt(6))^n-(5-2sqrt(6))^n]/[2sqrt(6)]. G.f.=2z/(1-10z+z^2). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 19 2005
%F A001078 a(-n)=-a(n).
%F A001078 a(n) = 9*(a(n-1)+a(n-2))-a(n-3). a(n) = 11*(a(n-1)-a(n-2))+a(n-3). - 
               Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 20 2006
%F A001078 a(n+1) = A054320(n) + A138288(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 12 2008
%F A001078 a(n) = Sinh[2n*ArcSinh[Sqrt[2]]]/Sqrt[6] - Herbert Kociemba (kociemba(AT)t-online.de), 
               Apr 24 2008
%F A001078 a(n)=2*A004189(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 26 2009]
%p A001078 A001078 := proc(n) option remember; if n=0 then 0 elif n=1 then 2 else 
               10*A001078(n-1)-A001078(n-2); fi; end;
%p A001078 A001078:=2*z/(1-10*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
%o A001078 (PARI) nxsqp1(m,n) = { for(x=1,m, y = n*x*x+1; if(issquare(y),print1(x" 
               ")) ) }
%o A001078 (PARI) a(n)=imag((5+2*quadgen(24))^n) /* Michael Somos Jul 05 2005 */
%o A001078 (PARI) a(n)=subst(poltchebi(n+1)-5*poltchebi(n),x,5)/12 /* Michael Somos 
               Jul 05 2005 */
%Y A001078 Cf. A053410.
%Y A001078 Cf. A138281.
%Y A001078 Sequence in context: A067641 A037566 A125857 this_sequence A001253 A085586 
               A136902
%Y A001078 Adjacent sequences: A001075 A001076 A001077 this_sequence A001079 A001080 
               A001081
%K A001078 nonn,easy,nice
%O A001078 0,2
%A A001078 N. J. A. Sloane (njas(AT)research.att.com).
%E A001078 Thanks to Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr) and Floor van 
               Lamoen (fvlamoen(AT)wxs.nl) for the Bottema references.

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