A221874 Numbers m such that 10*m^2 + 6 is a square.

1, 5, 43, 191, 1633, 7253, 62011, 275423, 2354785, 10458821, 89419819, 397159775, 3395598337, 15081612629, 128943316987, 572704120127, 4896450447169, 21747674952197, 185936173675435, 825838944063359, 7060678149219361, 31360132199455445

Offset: 1

Author

Bruno Berselli, Jan 28 2013

Keywords

• nonn
• easy

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).

Crossrefs

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

Subsequence of A031150.

Programs

Magma

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

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]

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);

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.