login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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