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A001078
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a(n) = 10*a(n-1) - a(n-2) with a(0) = 0, a(1) = 2.
(Formerly M2122 N0839)
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16
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0, 2, 20, 198, 1960, 19402, 192060, 1901198, 18819920, 186298002, 1844160100, 18255302998, 180708869880, 1788833395802, 17707625088140, 175287417485598, 1735166549767840, 17176378080192802, 170028614252160180, 1683109764441408998
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
Hacène Belbachir, Soumeya Merwa Tebtoub, 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.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (10,-1).
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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 *)
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CROSSREFS
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Cf. A053410.
Cf. A138281.
Sequence in context: A226312 A171076 A287999 * A299865 A001253 A303462
Adjacent sequences: A001075 A001076 A001077 * A001079 A001080 A001081
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Thanks to Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr) and Floor van Lamoen for the Bottema references.
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STATUS
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approved
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