A052454 -id:A052454 - cp's OEIS frontend

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

Bruno Berselli, Jan 28 2013

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.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

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

Subsequence of A031150.

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))));

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

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

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);

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.

A198943 y-values in the solution to 10*x^2-9 = y^2.

Original entry on oeis.org

1, 9, 41, 79, 351, 1559, 3001, 13329, 59201, 113959, 506151, 2248079, 4327441, 19220409, 85367801, 164328799, 729869391, 3241728359, 6240166921, 27715816449, 123100309841, 236962014199, 1052471155671, 4674570045599, 8998316372641, 39966188099049, 177510561422921

Offset: 1

Sture Sjöstedt, Oct 31 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..250
Index entries for linear recurrences with constant coefficients, signature (0,0,38,0,0,-1).

Cf. A052454.

LinearRecurrence[{0,0, 38, 0,0, -1}, {1, 9, 41, 79, 351, 1559}, 27]  (* Bruno Berselli, Nov 11 2011 *)

a(n+6) = 38*a(n+3)-a(n) with a(1)=1, a(2)=9, a(3)=41, a(4)=79, a(5)=351.

G.f.: x*(1+x)*(1+8*x+33*x^2+8*x^3+x^4)/(1-38*x^3+x^6). - Bruno Berselli, Nov 04 2011

More terms from Bruno Berselli, Nov 04 2011

A083000 Values of x for which 9y^2 = x^2 + 2xy - 2x has integer solutions with positive y.

Original entry on oeis.org

3, 18, 338, 2883, 24642, 486098, 4155987, 35532450, 700951682, 5992929075, 51237766962

Offset: 1

Jon Perry, May 30 2003

nonn

From the x and y values, one can derive some z's such that 10z^2-9 is a square (A052454): z = x+y-1. Other A052454 values can be derived from A046195.

Cf. A046195, A052454.

Edited by Don Reble, Nov 07 2005

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

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A198943 y-values in the solution to 10*x^2-9 = y^2.

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A083000 Values of x for which 9y^2 = x^2 + 2xy - 2x has integer solutions with positive y.

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