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 A075870 Numbers k such that 2*k^2 - 4 is a square. 15
 2, 10, 58, 338, 1970, 11482, 66922, 390050, 2273378, 13250218, 77227930, 450117362, 2623476242, 15290740090, 89120964298, 519435045698, 3027489309890, 17645500813642, 102845515571962, 599427592618130, 3493720040136818, 20362892648202778, 118683635849079850 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Lim_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). Also gives solutions to the equation x^2-2 = floor(x*r*floor(x/r)) where r=sqrt(2). - Benoit Cloitre, Feb 14 2004 The upper intermediate convergents to 2^(1/2) beginning with 10/7, 58/41, 338/239, 1970/1393 form a strictly decreasing sequence; essentially, numerators = A075870, denominators = A002315. - Clark Kimberling, Aug 27 2008 Numbers n such that sqrt(floor(n^2/2 - 1)) is an integer. The integer square roots are given by A002315. - Richard R. Forberg, Aug 01 2013 a(n) are the integer square roots of m^2 + (m+2)^2. The values of m are given by A065113 (except for m = 0). The values of this expression are given by A165518. - Richard R. Forberg, Aug 15 2013 Values of x (or y) in the solutions to x^2 - 6*x*y + y^2 + 16 = 0. - Colin Barker, Feb 04 2014 Also integers k such that k^2 is equal to the sum of four consecutive triangular numbers. - Colin Barker, Dec 20 2014 Equivalently, numbers x such that (x-1)*x/2 + x*(x+1)/2 = (y-1)^2 + (y+1)^2. y-values are listed in A002315. Example: for x=58 and y=41, 57*58/2 + 58*59/2 = 40^2 + 42^2. - Bruno Berselli, Mar 19 2018 REFERENCES A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966. L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400. Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Tanya Khovanova, Recursive Sequences J. J. O'Connor and E. F. Robertson, Pell's Equation Eric Weisstein's World of Mathematics, Pell Equation. Index entries for linear recurrences with constant coefficients, signature (6,-1). FORMULA a(n) = 2 * A001653(n). a(n) = (1/sqrt(2))*((1+sqrt(2))^(2*n-1) - (1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2). G.f.: 2*x*(1-x)/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008 a(n) = round(((2+sqrt(2))*(3+2*sqrt(2))^(n-1))/2). - Paul Weisenhorn, Jun 11 2020 EXAMPLE From Muniru A Asiru, Mar 19 2018: (Start) For k=2, 2*2^2 - 4 = 8 - 4 = 4 = 2^2. For k=10, 2*10^2 - 4 = 200 - 4 =  196 = 14^2. For k=58, 2*58^2 - 4 = 6728 - 4 =  6724 = 82^2. ... (End) MAPLE a:= proc(n) option remember: if n = 1 then 2 elif n = 2 then 10 elif  n >= 3 then 6*procname(n-1) - procname(n-2) fi; end: seq(a(n), n = 0..25); # Muniru A Asiru, Mar 19 2018 MATHEMATICA LinearRecurrence[{6, -1}, {2, 10}, 30] (* Harvey P. Dale, Sep 27 2018 *) PROG (PARI) Vec(2*x*(1-x)/(1-6*x+x^2) + O(x^100)) \\ Colin Barker, Dec 20 2014 (GAP) a:=[2, 10];; for n in [3..25] do a[n]:=6*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Mar 19 2018 CROSSREFS Cf. A000217, A000290, A002315. Twice A001653. Sequence in context: A235321 A248403 A278095 * A074608 A086871 A108450 Adjacent sequences:  A075867 A075868 A075869 * A075871 A075872 A075873 KEYWORD nonn,easy AUTHOR Gregory V. Richardson, Oct 16 2002 EXTENSIONS More terms from Colin Barker, Dec 20 2014 STATUS approved

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Last modified October 20 23:28 EDT 2020. Contains 337910 sequences. (Running on oeis4.)