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 A075796 Numbers k such that 5*k^2 + 5 is a square. 10
 2, 38, 682, 12238, 219602, 3940598, 70711162, 1268860318, 22768774562, 408569081798, 7331474697802, 131557975478638, 2360712083917682, 42361259535039638, 760141959546795802, 13640194012307284798, 244763350261984330562, 4392100110703410665318, 78813038642399407645162 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 Vincenzo Librandi, Table of n, a(n) for n = 1..200 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 (18,-1). FORMULA 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)). a(n) = 18*a(n-1) - a(n-2). a(n) = 2*A049629(n-1). Lim_{n->inf} a(n)/a(n-1) = 8*phi + 1 = 9 + 4*sqrt(5). a(n+1) = 9*a(n) + 4*sqrt(5)*sqrt((a(n)^2+1)). - Richard Choulet, Aug 30 2007 G.f.: 2*x*(1 + x)/(1 - 18*x + x^2). - Richard Choulet, Oct 09 2007 From Johannes W. Meijer, Jul 01 2010: (Start) a(n) = A000045(6*n+3) + A000045(6*n)/2. a(n) = 2*A167808(6*n+4) - A167808(6*n+6). Lim_{k->inf} a(n+k)/a(k) = A023039(n)*A060645(n)*sqrt(5). (End) 5*A007805(n)^2 - 1 = a(n+1)^2. - Sture Sjöstedt, Nov 29 2011 From Peter Bala, Nov 29 2013: (Start) a(n) = Lucas(6*n - 3)/2. Sum_{n >= 1} 1/(a(n) + 5/a(n)) = 1/4. Compare with A002878, A005248, A023039. (End) Lim_{n->inf} a(n)/A007805(n-1) = sqrt(5). - A.H.M. Smeets, May 29 2017 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 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 MAPLE with(combinat); A075796:=n->fibonacci(6*n+3)+fibonacci(6*n)/2; seq(A075796(n), n=1..50); # Wesley Ivan Hurt, Nov 29 2013 MATHEMATICA LinearRecurrence[{18, -1}, {2, 38}, 50] (* Sture Sjöstedt, Nov 29 2011; typo fixed by Vincenzo Librandi, Nov 30 2011 *) LucasL[6*Range[20]-3]/2 (* G. C. Greubel, Feb 13 2019 *) CoefficientList[Series[2*(1+x)/( 1-18*x+x^2 ), {x, 0, 20}], x] (* Stefano Spezia, Mar 02 2019 *) PROG (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 (MAGMA) [Lucas(6*n-3)/2: n in [1..20]]; // G. C. Greubel, Feb 13 2019 (PARI) vector(20, n, (fibonacci(6*n-2) + fibonacci(6*n-4))/2) \\ G. C. Greubel, Feb 13 2019 (Sage) [(fibonacci(6*n-2) + fibonacci(6*n-4))/2 for n in (1..20)] # G. C. Greubel, Feb 13 2019 CROSSREFS Cf. A000290, A306380. Sequence in context: A207320 A262585 A208240 * A230903 A246000 A266601 Adjacent sequences:  A075793 A075794 A075795 * A075797 A075798 A075799 KEYWORD nonn,easy AUTHOR Gregory V. Richardson, Oct 13 2002 STATUS approved

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Last modified April 5 03:58 EDT 2020. Contains 333238 sequences. (Running on oeis4.)