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%I #85 Feb 11 2024 04:37:06
%S 3,5,13,27,75,157,437,915,2547,5333,14845,31083,86523,181165,504293,
%T 1055907,2939235,6154277,17131117,35869755,99847467,209064253,
%U 581953685,1218515763,3391874643,7102030325,19769294173,41393666187
%N Numbers k such that (k^2 - 7)/2 is a square.
%C Lim_{n -> inf} a(n)/a(n-2) = 3 + 2*sqrt(2) = R1*R2. Lim_{k -> inf} a(2*k-1)/a(2*k) = (9 + 4*sqrt(2))/7 = R1 = A156649 (ratio #1). Lim_{k -> inf} a(2*k)/a(2*k-1) = (11 + 6*sqrt(2))/7 = R2 (ratio #2).
%C Also gives solutions > 3 to the equation x^2-4 = floor(x*r*floor(x/r)) where r=sqrt(2). - _Benoit Cloitre_, Feb 14 2004
%C From _Paul Curtz_, Dec 15 2012: (Start)
%C a(n-1) and A006452(n) are companions. Like A000129 and A001333.
%C Reduced mod 10 this is a sequence of period 12: 3, 5, 3, 7, 5, 7, 7, 5, 7, 3, 5, 3.
%C (End)
%C The Pisano periods (periods of the sequence reducing a(n) modulo m) for m>=1 are 1, 1, 8, 4, 12, 8, 6, 4, 24, 12, 24, 8, 28, ... _R. J. Mathar_, Dec 15 2012
%C Positive values of x (or y) satisfying x^2 - 6xy + y^2 + 56 = 0. - _Colin Barker_, Feb 08 2014
%C From _Wolfdieter Lang_, Feb 05 2015: (Start)
%C a(n+1) gives for n >= 0 all positive x solutions of the (generalized) Pell equation x^2 - 2*y^2 = +7.
%C The corresponding y solutions are given in A077442(n), n >= 0. The, e.g., the Nagell reference for finding all solutions.
%C Because the primitive Pythagorean triangle (3,4,5) is the only one with the sum of legs equal to 7 all positive solutions (x(n),y(n)) = (a(n+1),A077442(n)) of the Pell equation x^2 - 2*y^2 = +7 satisfy x(n) - y(n) < y(n) if n >= 1; only the first solution (x(0),y(0)) = (3,2) satisfies 3-1 > 1. Proof: Primitive Pythagorean triangles are characterized by the positive integer pairs [u,v] with u+v odd, gcd(u,v) = 1 and u > v. See the Niven et al. reference, Theorem 5.5, p. 232. The leg sum is L = (u+v)^2 - 2*v^2. With L = 7, x = u+v and y = v, every solution (x(n),y(n)) with x(n)-y(n) = u(n) > v(n) = y(n) will correspond to a primitive Pythagorean triangle. Note that because of gcd(x,y) = 1 also gcd(u,v) = 1. But there is only one such triangle with L=7, namely the one with [u(0),v(0)] = [2,1]. All other solutions with n >= 1 must therefore satisfy x(n)-y(n) < y(n). (End)
%C For n > 0, a(n+1) is the n-th almost Lucas-cobalancing number of first type (see Tekcan and Erdem). - _Stefano Spezia_, Nov 25 2022
%D 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.
%D L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
%D 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.
%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%D Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
%H Vincenzo Librandi, <a href="/A077443/b077443.txt">Table of n, a(n) for n = 1..1000</a>
%H Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://arxiv.org/abs/1810.07895">Classes of Gap Balancing Numbers</a>, arXiv:1810.07895 [math.NT], 2018.
%H Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://ajc.maths.uq.edu.au/pdf/77/ajc_v77_p318.pdf">Counting families of generalized balancing numbers</a>, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
%H J. J. O'Connor and E. F. Robertson, <a href="https://web.archive.org/web/20170729132724/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pell.html">History of Pell's Equation</a>
%H J. P. Robertson, <a href="https://web.archive.org/web/20180831180333/http://www.jpr2718.org/pell.pdf">Solving the Generalized Pell Equation</a>
%H Ahmet Tekcan and Alper Erdem, <a href="https://arxiv.org/abs/2211.08907">General Terms of All Almost Balancing Numbers of First and Second Type</a>, arXiv:2211.08907 [math.NT], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F a(2n+1) = A038762(n). a(2n) = A101386(n-1).
%F The same recurrences hold for the odd and the even indices: a(n+2) = 6*a(n) - a(n-2), a(n+1) = 3*a(n) + 2*(2*a(n)^2-14)^0.5 - _Richard Choulet_, Oct 11 2007
%F O.g.f.: -x*(x-1)*(3*x^2+8*x+3) / ( (x^2+2*x-1)*(x^2-2*x-1) ). - _R. J. Mathar_, Nov 23 2007
%F If n is even a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^-(1/2)*n) +(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^-(1/2)*n); if n is odd a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^((1/2)n-1/2)) +(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^((1/2)n-1/2)). - _Antonio Alberto Olivares_, Apr 20 2008
%F a(n) = A000129(n+1) + (-1)^n*A176981(n-1), n>1. - _R. J. Mathar_, Jul 03 2011
%F a(n) = A000129(n+1) -(-1)^n*A000129(n-2), rephrasing the formula above. - _Paul Curtz_, Dec 07 2012
%F a(n) = sqrt(8*A216134(n)^2 + 8*A216134(n) + 9) = 2*A124124(n) + 1. - _Raphie Frank_, May 24 2013
%F E.g.f.: cosh(sqrt(2)*x)*(3*cosh(x) - sinh(x)) + sqrt(2)*(2*cosh(x) - sinh(x))*sinh(sqrt(2)*x) - 3. - _Stefano Spezia_, Nov 25 2022
%e a(3)^2 - 2*A077442(2)^2 = 13^2 - 2*9^2 = +7. - _Wolfdieter Lang_, Feb 05 2015
%t LinearRecurrence[{0,6,0,-1},{3,5,13,27},50] (* _Sture Sjöstedt_, Oct 09 2012 *)
%Y Cf. A000129, A001333, A006452, A038761, A038762, A077442, A101386, A124124, A156649, A176981, A216134, A253811.
%K nonn,easy
%O 1,1
%A _Gregory V. Richardson_, Nov 06 2002
%E More terms from _Richard Choulet_, Oct 11 2007
%E Edited: replaced n by a(n) in the name. Moved Pell remarks to the comment section. Added cross references. - _Wolfdieter Lang_, Feb 05 2015
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