A159574 Positive numbers y such that y^2 is of the form x^2+(x+337)^2 with integer x.

313, 337, 365, 1513, 1685, 1877, 8765, 9773, 10897, 51077, 56953, 63505, 297697, 331945, 370133, 1735105, 1934717, 2157293, 10112933, 11276357, 12573625, 58942493, 65723425, 73284457, 343542025, 383064193, 427133117, 2002309657

Offset

1,1

Comments

(-25,a(1)) and (A129999(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+337)^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)=313, a(2)=337, a(3)=365, a(4)=1513, a(5)=1685, a(6)=1877.

G.f.: x*(1-x)*(313+650*x+1015*x^2+650*x^3+313*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 337*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) = (339+26*sqrt(2))/337 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 1.

Example

(-25, a(1)) = (-25, 313) is a solution: (-25)^2+(-25+337)^2 = 625+97344 = 97969 = 313^2.
(A129993(1), a(2)) = (0, 337) is a solution: 0^2+(0+337)^2 = 113569 = 337^2.
(A129993(3), a(4)) = (888, 1513) is a solution: 888^2+(888+337)^2 = 788544+1500625 = 2289169 = 1513^2.

Prog

(PARI) {forstep(n=-28, 50000000, [3, 1], if(issquare(2*n^2+674*n+113569, &k), print1(k, ", ")))}

Crossrefs

Cf. A129999, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576(decimal expansion of (278307+179662*sqrt(2))/337^2).

Sequence in context: A045275 A238057 A097023 * A139656 A061323 A082584

Adjacent sequences: A159571 A159572 A159573 * A159575 A159576 A159577

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 16 2009

Status

approved