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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

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Last modified April 15 00:44 EDT 2021. Contains 342971 sequences. (Running on oeis4.)