A001078 - OEIS

%I M2122 N0839 #97 May 16 2024 15:44:40

%S 0,2,20,198,1960,19402,192060,1901198,18819920,186298002,1844160100,

%T 18255302998,180708869880,1788833395802,17707625088140,

%U 175287417485598,1735166549767840,17176378080192802,170028614252160180,1683109764441408998

%N a(n) = 10*a(n-1) - a(n-2) with a(0) = 0, a(1) = 2.

%C Also 6*x^2+1 is a square. - _Cino Hilliard_, Mar 08 2003

%C 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 n such that 6*n^2 = floor(sqrt(6)*n*ceiling(sqrt(6)*n)). - _Benoit Cloitre_, May 10 2003

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 19 2005

%C (sqrt(2) + sqrt(3))^(2*n) = A001079(n) + a(n)*sqrt(6); a(n) = A054320(n) + A138288(n). - _Reinhard Zumkeller_, Mar 12 2008

%C Numbers m such that A000217(m) plus A000326(m) equals an octagonal number (A000567). For a(3)=198, A000217(198)=19701, A000326(198)=58707, therefore 19701 + 58707 = 78408 = A000567(162). - _Bruno Berselli_, Apr 15 2013

%D O. Bottema: Verscheidenheden XXVI. Het vraagstuk van Malfatti, Euclides 25 (1949-50), pp. 144-149 [in Dutch].

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 283, 302, P_{16}).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%D V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

%H G. C. Greubel, <a href="/A001078/b001078.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from T. D. Noe)

%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

%H O. Bottema, <a href="http://forumgeom.fau.edu/FG2001volume1/FG200107index.html">The Malfatti problem</a> (translation of Het vraagstuk van Malfatti), Forum Geom. 1 (2001) 43-50.

%H O. Bottema, <a href="/malfatti.html">Het Vraagstuk Van Malfatti</a>, from Euclides.

%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">De solutione problematum diophanteorum per numeros integros</a>, par. 18.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).

%F From _Emeric Deutsch_, Jun 19 2005: (Start)

%F a(n) = ((5 + 2*sqrt(6))^n - (5 - 2*sqrt(6))^n)/(2*sqrt(6)).

%F G.f.: 2*z/(1 - 10*z + z^2). (End)

%F a(-n) = -a(n).

%F From _Mohamed Bouhamida_, Sep 20 2006: (Start)

%F a(n) = 9*(a(n-1) + a(n-2)) - a(n-3).

%F a(n) = 11*(a(n-1) - a(n-2)) + a(n-3). (End)

%F a(n+1) = A054320(n) + A138288(n). - _Reinhard Zumkeller_, Mar 12 2008

%F a(n) = sinh(2n*arcsinh(sqrt(2)))/sqrt(6). - _Herbert Kociemba_, Apr 24 2008

%F a(n) = 2*A004189(n). - _R. J. Mathar_, Oct 26 2009

%F E.g.f.: 2*exp(5*x)*sinh(2*sqrt(6)*x)/(2*sqrt(6)). - _Stefano Spezia_, May 16 2023

%p 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:=2*z/(1-10*z+z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t a[0]=0; a[1]=2; a[n_] := a[n] = 10*a[n-1] - a[n-2]; Table[a[n],{n,0,19}] (* _Jean-François Alcover_, Mar 18 2011 *)

%t LinearRecurrence[{10,-1},{0,2},20] (* _Harvey P. Dale_, Jun 23 2011 *)

%t CoefficientList[Series[2*x/(1 - 10*x + x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 19 2017 *)

%o (PARI) nxsqp1(m,n) = { for(x=1,m, y = n*x*x+1; if(issquare(y),print1(x" ")) ) }

%o (PARI) a(n)=imag((5+2*quadgen(24))^n) /* _Michael Somos_, Jul 05 2005 */

%o (PARI) a(n)=subst(poltchebi(n+1)-5*poltchebi(n),x,5)/12 /* _Michael Somos_, Jul 05 2005 */

%o (Haskell)

%o a001078 n = a001078_list !! n

%o a001078_list =

%o    0 : 2 : zipWith (-) (map (10*) $ tail a001078_list) a001078_list

%o -- _Reinhard Zumkeller_, Mar 18 2011

%o (PARI) x='x+O('x^30); concat([0], Vec(2*x/(1 - 10*x + x^2))) \\ _G. C. Greubel_, Dec 19 2017

%o (Magma) I:=[0, 2]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 19 2017

%Y Cf. A000217, A000326, A000567, A001079, A004189, A053410, A054320, A138281, A138288.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Thanks to Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr) and _Floor van Lamoen_ for the Bottema references.