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A001084
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a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.
(Formerly M3167 N1284)
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3
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0, 3, 60, 1197, 23880, 476403, 9504180, 189607197, 3782639760, 75463188003, 1505481120300, 30034159217997, 599177703239640, 11953519905574803, 238471220408256420, 4757470888259553597, 94910946544782815520, 1893461460007396756803, 37774318253603152320540
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also 11*x^2+1 is a square. n=11 in PARI script below. - Cino Hilliard (hillcino368(AT)gmail.com), Mar 08 2003
a(n) solves for y in the Diophantine equation x^2-11*y^2=1,The corresponding x solutions are provided by A001085. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
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REFERENCES
| H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.
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).
"Questions D'Arithmetique", Question 3686, Solution by H.L. Mennessier, Mathesis, 65(4, Supplement) 1956, pp. 1-12.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| Lim a(n)/a(n-1) = 10 + 3*Sqrt(11); for all n in the sequence, 11*n^2 + 1 is a perfect square. - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002
a(n) = [(10+3*Sqrt(11))^n - (10-3*Sqrt(11))^n] / (2*Sqrt(11)) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002
a(n) = 19*(a(n-1)+a(n-2))-a(n-3). a(n) = 21*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 20 2006
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MAPLE
| A001084:=3*z/(1-20*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| LinearRecurrence[{20, -1}, {0, 3}, 20] (* T. D. Noe, Dec 19 2011 *)
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PROG
| (PARI) nxsqp1(m, n) = { for(x=1, m, y = n*x*x+1; if(issquare(y), print1(x" ")) ) }
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CROSSREFS
| Equals 3 * A075843. Cf. A001085
Sequence in context: A115490 A065889 A183251 * A137150 A081854 A085990
Adjacent sequences: A001081 A001082 A001083 * A001085 A001086 A001087
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
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