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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Crossrefs

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

Programs

Haskell

a001078 n = a001078_list !! n
a001078_list =
   0 : 2 : zipWith (-) (map (10*) $ tail a001078_list) a001078_list
-- Reinhard Zumkeller, Mar 18 2011

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

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 *)

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 */

PARI

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

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

Extensions

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