

A076218


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


8



0, 1, 5, 145, 4901, 166465, 5654885, 192099601, 6525731525, 221682772225, 7530688524101, 255821727047185, 8690408031080165, 295218051329678401, 10028723337177985445, 340681375412721826705, 11573138040695364122501, 393146012008229658338305
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Lim_{n > infinity} a(n)/a(n1) = 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 n1 and 2*n1 are perfect squares.  Sture Sjöstedt, Feb 22 2017


LINKS

Colin Barker, Table of n, a(n) for n = 1..650
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419427.
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))^(n1)) + (3+(1712*sqrt(2))^(n1)) )/8 for n>=1.
a(n) = 35 * a(n1)  35 * a(n2) + a(n3).
G.f.: (x30*x^2+5*x^3)/(135*x+35*x^2x^3). (End)
Product of adjacent oddindexed 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 evenindexed Pell numbers (A000129).  Charlie Marion, Mar 09 2005
a(n) = A001652(n1) + 2*A001652(n1)*A001652(n2) + A001652(n2) + 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*(130*x+5*x^2) / ((1x)*(134*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



