A001078 a(n) = 10*a(n-1) - a(n-2) with a(0) = 0, a(1) = 2. (Formerly M2122 N0839)

0, 2, 20, 198, 1960, 19402, 192060, 1901198, 18819920, 186298002, 1844160100, 18255302998, 180708869880, 1788833395802, 17707625088140, 175287417485598, 1735166549767840, 17176378080192802, 170028614252160180, 1683109764441408998

Offset

0,2

Comments

Also 6*x^2+1 is a square. - Cino Hilliard, Mar 08 2003

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

n such that 6*n^2 = floor(sqrt(6)*n*ceiling(sqrt(6)*n)). - Benoit Cloitre, May 10 2003

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 19 2005

(sqrt(2) + sqrt(3))^(2*n) = A001079(n) + a(n)*sqrt(6); a(n) = A054320(n) + A138288(n). - Reinhard Zumkeller, Mar 12 2008

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

References

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

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}).

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).

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

O. Bottema, The Malfatti problem (translation of Het vraagstuk van Malfatti), Forum Geom. 1 (2001) 43-50.

O. Bottema, Het Vraagstuk Van Malfatti, from Euclides.

L. Euler, De solutione problematum diophanteorum per numeros integros, par. 18.

Tanya Khovanova, Recursive Sequences.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Index entries for linear recurrences with constant coefficients, signature (10,-1).

Formula

From Emeric Deutsch, Jun 19 2005: (Start)

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

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

a(-n) = -a(n).

From Mohamed Bouhamida, Sep 20 2006: (Start)

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

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

a(n+1) = A054320(n) + A138288(n). - Reinhard Zumkeller, Mar 12 2008

a(n) = sinh(2n*arcsinh(sqrt(2)))/sqrt(6). - Herbert Kociemba, Apr 24 2008

a(n) = 2*A004189(n). - R. J. Mathar, Oct 26 2009

E.g.f.: 2*exp(5*x)*sinh(2*sqrt(6)*x)/(2*sqrt(6)). - Stefano Spezia, May 16 2023

Maple

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;
A001078:=2*z/(1-10*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
LinearRecurrence[{10, -1}, {0, 2}, 20] (* Harvey P. Dale, Jun 23 2011 *)
CoefficientList[Series[2*x/(1 - 10*x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

Prog

(PARI) nxsqp1(m, n) = { for(x=1, m, y = n*x*x+1; if(issquare(y), print1(x" ")) ) }
(PARI) a(n)=imag((5+2*quadgen(24))^n) /* Michael Somos, Jul 05 2005 */
(PARI) a(n)=subst(poltchebi(n+1)-5*poltchebi(n), x, 5)/12 /* Michael Somos, Jul 05 2005 */
(Haskell)
a001078 n = a001078_list !! n
a001078_list =
   0 : 2 : zipWith (-) (map (10*) $ tail a001078_list) a001078_list
-- Reinhard Zumkeller, Mar 18 2011
(PARI) x='x+O('x^30); concat([0], Vec(2*x/(1 - 10*x + x^2))) \\ G. C. Greubel, Dec 19 2017
(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

Crossrefs

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

Sequence in context: A171076 A287999 A348886 * A299865 A001253 A303462

Adjacent sequences: A001075 A001076 A001077 * A001079 A001080 A001081

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Thanks to Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr) and Floor van Lamoen for the Bottema references.

Status

approved