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A075796 Numbers k such that 5*k^2 + 5 is a square.

%I A075796 #105 Jul 23 2025 14:42:52
%S A075796 2,38,682,12238,219602,3940598,70711162,1268860318,22768774562,
%T A075796 408569081798,7331474697802,131557975478638,2360712083917682,
%U A075796 42361259535039638,760141959546795802,13640194012307284798,244763350261984330562,4392100110703410665318,78813038642399407645162
%N A075796 Numbers k such that 5*k^2 + 5 is a square.
%C A075796 Bisection of A001077; a(n) = A001077(2*n-1). - _Greg Dresden_, Jun 08 2021
%C A075796 From _Peter Bala_, Aug 25 2022: (Start)
%C A075796 The aerated sequence (b(n))n>=1 = [2, 0, 38, 0, 682, 0, 1238, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). The sequence (1/2)*(b(n))n>=1 is the case P1 = 0, P2 = -16, Q = -1  of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. (End)
%D A075796 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 A075796 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
%D A075796 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.
%H A075796 Vincenzo Librandi, <a href="/A075796/b075796.txt">Table of n, a(n) for n = 1..200</a>
%H A075796 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A075796 J. J. O'Connor and E. F. Robertson, <a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pell/">Pell's Equation</a>
%H A075796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation.</a>
%H A075796 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A075796 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).
%F A075796 a(n) = (((9 + 4*sqrt(5))^n - (9 - 4*sqrt(5))^n) + ((9 + 4*sqrt(5))^(n-1) - (9 - 4*sqrt(5))^(n-1)))/(4*sqrt(5)).
%F A075796 a(n) = 18*a(n-1) - a(n-2).
%F A075796 a(n) = 2*A049629(n-1).
%F A075796 Limit_{n->oo} a(n)/a(n-1) = 8*phi + 1 = 9 + 4*sqrt(5).
%F A075796 a(n+1) = 9*a(n) + 4*sqrt(5)*sqrt((a(n)^2+1)). - _Richard Choulet_, Aug 30 2007
%F A075796 G.f.: 2*x*(1 + x)/(1 - 18*x + x^2). - _Richard Choulet_, Oct 09 2007
%F A075796 From _Johannes W. Meijer_, Jul 01 2010: (Start)
%F A075796 a(n) = A000045(6*n+3) + A000045(6*n)/2.
%F A075796 a(n) = 2*A167808(6*n+4) - A167808(6*n+6).
%F A075796 Limit_{k->oo} a(n+k)/a(k) = A023039(n)*A060645(n)*sqrt(5).
%F A075796 (End)
%F A075796 5*A007805(n)^2 - 1 = a(n+1)^2. - _Sture Sjöstedt_, Nov 29 2011
%F A075796 From _Peter Bala_, Nov 29 2013: (Start)
%F A075796 a(n) = Lucas(6*n - 3)/2.
%F A075796 Sum_{n >= 1} 1/(a(n) + 5/a(n)) = 1/4. Compare with A002878, A005248, A023039. (End)
%F A075796 Limit_{n->oo} a(n)/A007805(n-1) = sqrt(5). - _A.H.M. Smeets_, May 29 2017
%F A075796 E.g.f.: (exp((9 - 4*sqrt(5))*x)*(- 5 + 2*sqrt(5) + (5 + 2*sqrt(5))*exp(8*sqrt(5)*x)))/(2*sqrt(5)). - _Stefano Spezia_, Feb 13 2019
%F A075796 Sum_{n > 0} 1/a(n) = (1/log(9 - 4*sqrt(5)))*(- 17 - 38/sqrt(5))*sqrt(5*(9 - 4*sqrt(5)))*(- 9 + 4*sqrt(5))*(psi_{9 - 4*sqrt(5)}(1/2) - psi_{9 - 4*sqrt(5)}(1/2 - (I*Pi)/log(9 - 4*sqrt(5)))) approximately equal to 0.527868600269500798938265500122302016..., where psi_q(x) is the q-digamma function. - _Stefano Spezia_, Feb 25 2019
%F A075796 a(n) = sinh((6*n - 3)*arccsch(2)). - _Peter Luschny_, May 25 2022
%p A075796 with(combinat); A075796:=n->fibonacci(6*n+3)+fibonacci(6*n)/2; seq(A075796(n), n=1..50); # _Wesley Ivan Hurt_, Nov 29 2013
%t A075796 LinearRecurrence[{18, -1}, {2, 38}, 50] (* _Sture Sjöstedt_, Nov 29 2011; typo fixed by _Vincenzo Librandi_, Nov 30 2011 *)
%t A075796 LucasL[6*Range[20]-3]/2 (* _G. C. Greubel_, Feb 13 2019 *)
%t A075796 CoefficientList[Series[2*(1+x)/( 1-18*x+x^2 ), {x,0,20}],x] (* _Stefano Spezia_, Mar 02 2019 *)
%o A075796 (Magma) I:=[2,38]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 30 2011
%o A075796 (Magma) [Lucas(6*n-3)/2: n in [1..20]]; // _G. C. Greubel_, Feb 13 2019
%o A075796 (PARI) vector(20, n, (fibonacci(6*n-2) + fibonacci(6*n-4))/2) \\ _G. C. Greubel_, Feb 13 2019
%o A075796 (Sage) [(fibonacci(6*n-2) + fibonacci(6*n-4))/2 for n in (1..20)] # _G. C. Greubel_, Feb 13 2019
%Y A075796 Cf. A000290, A306380, A001077.
%K A075796 nonn,easy
%O A075796 1,1
%A A075796 _Gregory V. Richardson_, Oct 13 2002