A309998 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 529)^2 = y^2.

0, 276, 287, 740, 759, 1587, 3059, 3120, 5687, 5796, 10580, 19136, 19491, 34440, 35075, 62951, 112815, 114884, 202011, 205712, 368184, 658812, 670871, 1178684, 1200255, 2147211, 3841115, 3911400, 6871151, 6996876, 12516140, 22388936, 22798587, 40049280, 40782059, 72950687

Offset: 1

Author

Mohamed Bouhamida, Aug 26 2019

Keywords

• nonn
• easy

Comments

For the generic case x^2 + (x + p^2)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m>=5, (0; p^2) and (2*m^3 + 2*m^2 - 4*m - 4; m^4 + 2*m^3 - 4*m - 4) are solutions.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,6,-6,0,0,0,-1,1).

Crossrefs

Cf. A207059.

Programs

Mathematica

Rest@ CoefficientList[Series[x^2*(276 + 11 x + 453 x^2 + 19 x^3 + 828 x^4 - 184 x^5 - 5 x^6 - 151 x^7 - 5 x^8 - 184 x^9)/((1 - x) (1 - 6 x^5 + x^10)), {x, 0, 36}], x] (* Michael De Vlieger, Sep 29 2019 *)

PARI

concat(0, Vec(x^2*(276 + 11*x + 453*x^2 + 19*x^3 + 828*x^4 - 184*x^5 - 5*x^6 - 151*x^7 - 5*x^8 - 184*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^40))) \\ Colin Barker, Aug 27 2019

Formula

a(n) = 6*a(n-5) - a(n-10) + 1058 with a(0) = 0, a(1) = 276, a(2) = 287, a(3) = 740, a(4) = 759, a(5) = 1587, a(6) = 3059, a(7) = 3120, a(8) = 5687, a(9) = 5796.

From Colin Barker, Aug 27 2019: (Start)

G.f.: x^2*(276 + 11*x + 453*x^2 + 19*x^3 + 828*x^4 - 184*x^5 - 5*x^6 - 151*x^7 - 5*x^8 - 184*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).

a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.

(End)