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A011922
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a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).
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9
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1, 3, 33, 451, 6273, 87363, 1216801, 16947843, 236052993, 3287794051, 45793063713, 637815097923, 8883618307201, 123732841202883, 1723376158533153, 24003533378261251, 334326091137124353
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history;
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internal format)
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OFFSET
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0,2
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REFERENCES
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Mario Velucchi, Seeing couples, in Recreational and Educational Computing, to appear 1997.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..100
Christian Aebi, and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Z. Franusic, On the Extension of the Diophantine Pair {1,3} in Z[surd d], J. Int. Seq. 13 (2010) # 10.9.6.
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
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FORMULA
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a(n) = (2+sqrt(1+((((2+sqrt(3))^(2*n)-(2-sqrt(3))^(2*n))^2)/4)))/3.
a(n) = ((7+4*sqrt(3))^n+(7-4*sqrt(3))^n+4)/6. - Bruno Berselli, Jul 09 2011
G.f.: (1-12*x+3*x^2)/ ((1-x) * (x^2-14*x+1)). - R. J. Mathar, Apr 15 2010
Sqrt(3) = 1 + sum(n>=1, 2/a(n)) = 1 + 2/3 + 2/33 +... - Gary W. Adamson, Jun 12 2003
a(n)^2 = A103974(n+1)^2 - (4*A007655(n+1))^2. - Paul D. Hanna, Mar 06 2005
a(n) = (A011943(n+1) + 2)/3. - Ralf Stephan, Aug 13 2013
a(n) = A001075(n)^2 - A001353(n)^2. - Richard R. Forberg, Aug 24 2013
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MAPLE
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a:= gfun:-rectoproc({a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3), a(0)=1, a(1)=3, a(2)=33}, a(n), remember):
map(a, [$0..100]); # Robert Israel, Jul 02 2015
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MATHEMATICA
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RecurrenceTable[{a[n] == 15 a[n - 1] - 15 a[n - 2] + a[n - 3], a[0] == 1, a[1] == 3, a[2] == 33}, a, {n, 0, 15}] (* Michael De Vlieger, Jul 02 2015 *)
LinearRecurrence[{15, -15, 1}, {1, 3, 33}, 30] (* Harvey P. Dale, Dec 04 2018 *)
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PROG
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(Maxima) a[0]:1$ a[1]:3$ a[2]:33$ a[n]:=15*a[n-1]-15*a[n-2]+a[n-3]$ makelist(a[n], n, 0, 16); \\ Bruno Berselli, Jul 09 2011
(MAGMA) I:=[1, 3, 33]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..17]]; // Bruno Berselli, Jul 09 2011
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -15, 15]^n*[1; 3; 33])[1, 1] \\ Charles R Greathouse IV, Jul 02 2015
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CROSSREFS
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Cf. A011916, A011918, A011920, A103974, A007655.
Sequence in context: A009502 A222941 A321265 * A264833 A071405 A234526
Adjacent sequences: A011919 A011920 A011921 * A011923 A011924 A011925
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KEYWORD
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nonn,easy
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AUTHOR
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Mario Velucchi (mathchess(AT)velucchi.it)
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EXTENSIONS
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Formula corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001
Recurrence in definition by R. J. Mathar, Apr 15 2010
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STATUS
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approved
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