A076218 Numbers n such that 2*n^2 - 3*n + 1 is a square.

0, 1, 5, 145, 4901, 166465, 5654885, 192099601, 6525731525, 221682772225, 7530688524101, 255821727047185, 8690408031080165, 295218051329678401, 10028723337177985445, 340681375412721826705, 11573138040695364122501, 393146012008229658338305

Offset

1,3

Comments

Limit_{n -> infinity} a(n)/a(n-1) = 33.970562748477140585620264690516... = 17 + 12*sqrt(2).

Conjecture: a nonzero number occurs twice in A055524 if and only if it's in this sequence. - J. Lowell, Jul 23 2016

Equivalently, n=0 or both n-1 and 2*n-1 are perfect squares. - Sture Sjöstedt, Feb 22 2017

Links

Colin Barker, Table of n, a(n) for n = 1..650

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

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 (35,-35,1).

Formula

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 04 2002: (Start)

a(n) = ( (3+(17+12*sqrt(2))^(n-1)) + (3+(17-12*sqrt(2))^(n-1)) )/8 for n>=1.

a(n) = 35 * a(n-1) - 35 * a(n-2) + a(n-3).

G.f.: (x-30*x^2+5*x^3)/(1-35*x+35*x^2-x^3). (End)

Product of adjacent odd-indexed Pell numbers (A000129). - Gary W. Adamson, Jun 07 2003

sqrt(2) - 1 = 0.414213562... = 2/5 + 2/145 + 2/4901 + 2/166465 + ... = Sum_{n>=2} 2/a(n). - Gary W. Adamson, Jun 07 2003

For n > 0, one more than square of adjacent even-indexed Pell numbers (A000129). - Charlie Marion, Mar 09 2005

a(n) = A001652(n-1) + 2*A001652(n-1)*A001652(n-2) + A001652(n-2) + 2. - Charlie Marion, Nov 24 2018

Example

5 is in the sequence since 2*5^2 - 3*5 + 1 = 50 - 15 + 1 = 36 is a square. - Michael B. Porter, Jul 24 2016

Mathematica

Join[{0}, LinearRecurrence[{35, -35, 1}, {1, 5, 145}, 20]] (* Harvey P. Dale, Nov 27 2012 *)

Prog

(PARI) a(n)=if(n>1, ([0, 1, 0; 0, 0, 1; 1, -35, 35]^n*[145; 5; 1])[1, 1], 0) \\ Charles R Greathouse IV, Jul 24 2016
(PARI) concat(0, Vec(x^2*(1-30*x+5*x^2) / ((1-x)*(1-34*x+x^2)) + O(x^30))) \\ Colin Barker, Nov 21 2016

Crossrefs

Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: A084703 (k=-1), this sequence (k=3), A278310 (k=-5).

Sequence in context: A322954 A254711 A273920 * A267989 A307902 A281427

Adjacent sequences: A076215 A076216 A076217 * A076219 A076220 A076221

Keyword

nonn,easy

Author

Gregory V. Richardson, Nov 03 2002

Status

approved