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A076218 Numbers n such that 2*n^2 - 3*n + 1 is a square. 9
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
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
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.
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified June 16 11:02 EDT 2024. Contains 373429 sequences. (Running on oeis4.)