A221763 Numbers m such that 11*m^2 - 7 is a square.

1, 4, 16, 79, 319, 1576, 6364, 31441, 126961, 627244, 2532856, 12513439, 50530159, 249641536, 1008070324, 4980317281, 20110876321, 99356704084, 401209456096, 1982153764399, 8004078245599, 39543718583896, 159680355455884, 788892217913521

Offset: 1

Author

Bruno Berselli, Jan 24 2013

Keywords

• nonn
• easy

Comments

See the first comment of A221762.

a(n) == 1 (mod 3).

a(n+1)/a(n) tends alternately to (2+sqrt(11))^2/7 and (5+sqrt(11))^2/14; a(n+2)/a(n) tends to A176395^2/2.

Positive values of x (or y) satisfying x^2 - 20xy + y^2 + 63 = 0. - Colin Barker, Feb 18 2014

Links

• Bruno Berselli, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

Crossrefs

Cf. A221762.

Programs

Magma

m:=24; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+4*x-4*x^2-x^3)/(1-20*x^2+x^4)));

Magma

I:=[1,4,16,79]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013

Maple

A221763:=proc(q)
local n;
for n from 1 to q do if type(sqrt(11*n^2-7), integer) then print(n);
fi; od; end:
A221763(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

LinearRecurrence[{0, 20, 0, -1}, {1, 4, 16, 79}, 24]
CoefficientList[Series[(1 + 4 x - 4 x^2 - x^3)/(1 - 20 x^2 + x^4), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Maxima

makelist(expand(((11+2*sqrt(11)*(-1)^n)*(10-3*sqrt(11))^floor(n/2)+(11-2*sqrt(11)*(-1)^n)*(10+3*sqrt(11))^floor(n/2))/22), n, 1, 24);

Formula

G.f.: x*(1+4*x-4*x^2-x^3)/(1-20*x^2+x^4).

a(n) = ((11+2*t*(-1)^n)*(10-3*t)^floor(n/2)+(11-2*t*(-1)^n)*(10+3*t)^floor(n/2))/22, where t=sqrt(11).

a(n)*a(n-3)-a(n-1)*a(n-2) = (3/2)*(9+(-1)^n).