login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A221874 Numbers m such that 10*m^2 + 6 is a square. 7
1, 5, 43, 191, 1633, 7253, 62011, 275423, 2354785, 10458821, 89419819, 397159775, 3395598337, 15081612629, 128943316987, 572704120127, 4896450447169, 21747674952197, 185936173675435, 825838944063359, 7060678149219361, 31360132199455445 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The Diophantine equation 10*x^2+k = y^2, for |k|<10, has integer solutions with the following k values:

k = 1, the nonnegative x values are in A084070;

k = -1, " A097315;

k = 4, " 2*A084070;

k = -4, " 2*A097315;

k = 6, " this sequence;

k = -6, " A221875;

k = 9, " A075836;

k = -9, " A052454.

a(n+1)/a(n) tends alternately to (sqrt(2)+sqrt(5))^2/3 and (2*sqrt(2)+sqrt(5))^2/3; a(n+2)/a(n) tends to A176398^2.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

FORMULA

G.f.: x*(1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2)).

a(n) = ((-5*(-1)^n+2*t)*(3+t)^(2*floor(n/2))-(5*(-1)^n+2*t)*(3-t)^(2*floor(n/2)))/10, where t=sqrt(10).

a(n) = 2*A129556(n)+1.

a(n)*a(n-3)-a(n-1)*a(n-2) = -36+12(-1)^n.

MAPLE

A221874:=proc(q)

local n;

for n from 1 to q do if type(sqrt(10*n^2+6), integer) then print(n);

fi; od; end:

A221874(100000000000000000); # Paolo P. Lava, Feb 11 2013

MATHEMATICA

LinearRecurrence[{0, 38, 0, -1}, {1, 5, 43, 191}, 22]

PROG

(Magma) m:=22; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2))));

(Maxima) makelist(expand(((-5*(-1)^n+2*sqrt(10))*(3+sqrt(10))^(2*floor(n/2))-(5*(-1)^n+2*sqrt(10))*(3-sqrt(10))^(2*floor(n/2)))/10), n, 1, 22);

CROSSREFS

Cf. A097315, A084070, A075836, A052454, A129556, A221875.

Subsequence of A031150.

Sequence in context: A102851 A173554 A126963 * A317282 A182191 A038140

Adjacent sequences: A221871 A221872 A221873 * A221875 A221876 A221877

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jan 28 2013

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 14:20 EST 2022. Contains 358656 sequences. (Running on oeis4.)