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, and 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) = (A011943(n+1) + 2)/3. - Ralf Stephan, Aug 13 2013
E.g.f.: exp(x)*(2 + exp(6*x)*cosh(4*sqrt(3)*x))/3. - Stefano Spezia, Dec 11 2022
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
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