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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 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

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