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A221762 Numbers m such that 11*m^2 + 5 is a square. 6
1, 2, 22, 41, 439, 818, 8758, 16319, 174721, 325562, 3485662, 6494921, 69538519, 129572858, 1387284718, 2584962239, 27676155841, 51569671922, 552135832102, 1028808476201, 11015040486199, 20524599852098, 219748673891878, 409463188565759 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Corresponding squares are: 16, 49, 5329, 18496, 2119936, 7360369, 843728209, 2929407376, ... (subsequence of A016778).

The Diophantine equation 11*x^2+k = y^2, for |k|<11, has integer solutions with the following k values:

k = -10, the nonnegative x values are in A198947;

k =  -8,            "                    2*A075839;

k =  -7,            "                    A221763;

k =  -2,            "                    A075839;

k =   1,            "                    A001084;

k =   4,            "                    A075844;

k =   5,            "                    this sequence;

k =   9,            "                    3*A001084.

Also, the Diophantine equation h*x^2+5 = y^2 has infinitely many integer solutions for h = 5, 11, 19, 20, 29, 31, 41, 44, 55, 59, ...

a(n+1)/a(n) tends alternately to (1+sqrt(11))^2/10 and (4+sqrt(11))^2/5.

a(n+2)/a(n) tends to A176395^2/2.

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).

FORMULA

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

a(n) = -a(1-n) = ((-11*(-1)^n+4*t)*(10+3*t)^floor(n/2)-(11*(-1)^n+4*t)*(10-3*t)^floor(n/2))/22, where t=sqrt(11).

a(n) = 20*a(n-2) - a(n-4) for n>4, a(1)=1, a(2)=2, a(3)=22, a(4)=41.

a(n)*a(n-3)-a(n-1)*a(n-2) = -(3/2)*(9-7*(-1)^n).

a(n+1) + a(n-1) =  A198949(n), with a(0)=-1.

2*a(n-1) - a(n) =  A001084(n/2-1) for even n.

MAPLE

A221762:=proc(q)

local n;

for n from 1 to q do if type(sqrt(11*n^2+5), integer) then print(n);

fi; od; end:

A221762(1000); # Paolo P. Lava, Feb 19 2013

MATHEMATICA

LinearRecurrence[{0, 20, 0, -1}, {1, 2, 22, 41}, 24]

CoefficientList[Series[(1 + 2 x + 2 x^2 + x^3)/(1 - 20 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 18 2013 *)

PROG

(Maxima) makelist(expand(((-11*(-1)^n+4*sqrt(11))*(10+3*sqrt(11))^floor(n/2)-(11*(-1)^n+4*sqrt(11))*(10-3*sqrt(11))^floor(n/2))/22), n, 1, 24);

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

(MAGMA) I:=[1, 2, 22, 41]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 18 2013

CROSSREFS

Cf. A001084, A075839, A075844, A198947, A198949, A221763.

Cf. A049629 (numbers m such that 20*m^2 + 5 is a square), A075796 (numbers m such that 5*m^2 + 5 is a square).

Sequence in context: A126913 A019593 A060108 * A154798 A080142 A306969

Adjacent sequences:  A221759 A221760 A221761 * A221763 A221764 A221765

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jan 24 2013

STATUS

approved

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Last modified October 20 09:24 EDT 2021. Contains 348100 sequences. (Running on oeis4.)