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 A011922 a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3). 9
 1, 3, 33, 451, 6273, 87363, 1216801, 16947843, 236052993, 3287794051, 45793063713, 637815097923, 8883618307201, 123732841202883, 1723376158533153, 24003533378261251, 334326091137124353 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES Mario Velucchi, Seeing couples, in Recreational and Educational Computing, to appear 1997. LINKS 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). FORMULA 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 MAPLE 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 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A011916, A011918, A011920, A103974, A007655. Sequence in context: A009502 A222941 A321265 * A264833 A071405 A234526 Adjacent sequences:  A011919 A011920 A011921 * A011923 A011924 A011925 KEYWORD nonn,easy AUTHOR Mario Velucchi (mathchess(AT)velucchi.it) EXTENSIONS Formula corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001 Recurrence in definition by R. J. Mathar, Apr 15 2010 STATUS approved

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Last modified May 18 10:19 EDT 2021. Contains 343995 sequences. (Running on oeis4.)