A296377 Natural numbers y such that 7y^2 = x^2 + x + 1 has a solution in natural numbers.

1, 7, 247, 1777, 62737, 451351, 15934951, 114641377, 4047414817, 29118458407, 1028027428567, 7395973794001, 261114919441201, 1878548225217847, 66322161510636487, 477143853231539137, 16845567908782226497, 121192660172585722951, 4278707926669174893751

Offset

1,2

Comments

Given explicitly as the denominators of the convergents to the continued fractions

[2,(1,1,1,4)^i,5,(1,1,1,4)^{i-1},1,2] (for n odd and i = (n-1)/2)

and

[2,(1,1,1,4)^i,1,1,2,(1,4,1,1)^i,1] (for n even and i = n/2 - 1).

References

E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - N. J. A. Sloane, Mar 02 2022

Links

Colin Barker, Table of n, a(n) for n = 1..832

Index entries for linear recurrences with constant coefficients, signature (0,254,0,-1).

Formula

Recurrence: a(n) = 255*a(n-2) - 255*a(n-4) + a(n-6).

From Colin Barker, Dec 11 2017: (Start)

G.f.: x*(1-x)*(1+8*x+x^2) / ((1-16*x+x^2)*(1+16*x+x^2)).

a(n) = 254*a(n-2) - a(n-4) for n>4.

(End)

Example

For n = 3 the pair is (x,y) = (653,247).

Prog

(PARI) Vec(x*(1-x)*(1+8*x+x^2) / ((1-16*x+x^2)*(1+16*x+x^2)) + O(x^30)) \\ Colin Barker, Dec 13 2017

Crossrefs

Cf. A296376.

Sequence in context: A330517 A086214 A133589 * A223630 A203158 A227414

Adjacent sequences: A296374 A296375 A296376 * A296378 A296379 A296380

Keyword

nonn,easy

Author

Jeffrey Shallit, Dec 11 2017

Status

approved