A296376 Natural numbers x such that 7*y^2 = x^2 + x + 1 has a solution in natural numbers.

2, 18, 653, 4701, 165986, 1194162, 42159917, 303312573, 10708453058, 77040199506, 2719904916941, 19567907362077, 690845140450082, 4970171429768178, 175471945769404013, 1262403975253755261, 44569183380288169346, 320645639543024068242, 11320397106647425609997, 81442730039952859578333

Offset

1,1

Comments

Given explicitly as the numerators 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).

a(n) == 10 + 8*(-1)^n (mod 21). - Robert Israel, Dec 13 2017

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..800

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

Formula

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

From Colin Barker, Dec 13 2017: (Start)

G.f.: x*(2 + 16*x + 127*x^2 - 16*x^3 - 3*x^4) / ((1 - x)*(1 - 16*x + x^2)*(1 + 16*x + x^2)).

a(n) = a(n-1) + 254*a(n-2) - 254*a(n-3) - a(n-4) + a(n-5) for n>5.

(End)

4*a(n) = 7*( -A077412(n) +17*A077412(n-1) ) -3*( (-1)^n*A077412(n) -15*(-1)^n*A077412(n-1) ) - 2 . - R. J. Mathar, Mar 07 2022

Example

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

Maple

f:= gfun:-rectoproc({a(n) = 255*a(n-2) - 255*a(n-4) + a(n-6), a(1)=2, a(2)=18, a(3)=653, a(4)=4701, a(5)= 165986, a(6)=1194162}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Dec 12 2017

Prog

(PARI) Vec(x*(2 + 16*x + 127*x^2 - 16*x^3 - 3*x^4) / ((1 - x)*(1 - 16*x + x^2)*(1 + 16*x + x^2)) + O(x^25)) \\ Colin Barker, Dec 13 2017

Crossrefs

Cf. A296377.

Sequence in context: A324308 A071352 A258384 * A013035 A350008 A132520

Adjacent sequences: A296373 A296374 A296375 * A296377 A296378 A296379

Keyword

nonn,easy

Author

Jeffrey Shallit, Dec 11 2017

Extensions

More terms from Robert Israel, Dec 12 2017

Status

approved