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A075796 Numbers k such that 5*k^2 + 5 is a square. 8
2, 38, 682, 12238, 219602, 3940598, 70711162, 1268860318, 22768774562, 408569081798, 7331474697802, 131557975478638, 2360712083917682, 42361259535039638, 760141959546795802, 13640194012307284798 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Lim_{n->inf} a(n)/a(n-1) = 8*phi + 1 = 9 + 4*sqrt(5).

Lim_{n->inf} a(n)/A007805(n-1) = sqrt(5). - A.H.M. Smeets, May 29 2017

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).

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

a(n) = 1/2*Lucas(6*n - 3). Sum_{n >= 1} 1/( a(n) + 5/a(n) ) = 1/4. Compare with A002878, A005248, A023039. - Peter Bala, Nov 29 2013

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 *)

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

CROSSREFS

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 October 23 20:10 EDT 2017. Contains 293813 sequences.