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A159574 Positive numbers y such that y^2 is of the form x^2+(x+337)^2 with integer x. 3
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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

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

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.: (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.

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

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Last modified October 14 05:04 EDT 2019. Contains 327995 sequences. (Running on oeis4.)