A157348 Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x.

229, 281, 365, 1009, 1405, 1961, 5825, 8149, 11401, 33941, 47489, 66445, 197821, 276785, 387269, 1152985, 1613221, 2257169, 6720089, 9402541, 13155745, 39167549, 54802025, 76677301, 228285205, 319409609, 446908061, 1330543681

Offset

1,1

Comments

(-60, a(1)) and (A129626(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

Links

Table of n, a(n) for n=1..28.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=229, a(2)=281, a(3)=365, a(4)=1009, a(5)=1405, a(6)=1961.

G.f.: x*(1-x)*(229+510*x+875*x^2+510*x^3+229*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 281*A001653(k) for k >= 1.

Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).

Limit_{n -> oo} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 1.

Example

(-60, a(1)) = (-60, 229) is a solution: (-60)^2+(-60+281)^2 = 3600+48841 = 52441 = 229^2.
(A129626(1), a(2)) = (0, 281) is a solution: 0^2+(0+281)^2 = 78961 = 281^2.
(A129626(3), a(4)) = (559, 1009) is a solution: 559^2+(559+281)^2 = 312481+705600 = 1018081 = 1009^2.

Prog

(PARI) {forstep(n=-60, 200000000, [3, 1], if(issquare(2*n^2+562*n+78961, &k), print1(k, ", ")))}

Crossrefs

Cf. A129626, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

Sequence in context: A250237 A350165 A112847 * A142221 A142779 A139512

Adjacent sequences: A157345 A157346 A157347 * A157349 A157350 A157351

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 12 2009

Status

approved