A011922 a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).

1, 3, 33, 451, 6273, 87363, 1216801, 16947843, 236052993, 3287794051, 45793063713, 637815097923, 8883618307201, 123732841202883, 1723376158533153, 24003533378261251, 334326091137124353, 4656561742541479683, 64857538304443591201, 903348974519668797123, 12582028104970919568513

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)^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

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

Cf. A001353, A001075, A007655, A011916, A011918, A011920, A011943, A103974.

Sequence in context: A009502 A222941 A321265 * A365028 A368442 A264833

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