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A160574 Positive numbers y such that y^2 is of the form x^2+(x+313)^2 with integer x. 3
233, 313, 493, 905, 1565, 2725, 5197, 9077, 15857, 30277, 52897, 92417, 176465, 308305, 538645, 1028513, 1796933, 3139453, 5994613, 10473293, 18298073, 34939165, 61042825, 106648985, 203640377, 355783657, 621595837, 1186903097 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-105, a(1)) and (A129640(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 1.

LINKS

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

FORMULA

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=233, a(2)=313, a(3)=493, a(4)=905, a(5)=1565, a(6)=2725.

G.f.: (1-x)*(233+546*x+1039*x^2+546*x^3+233*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 313*A001653(k) for k >= 1.

EXAMPLE

(-105, a(1)) = (-105, 233) is a solution: (-105)^2+(-105+313)^2 = 11025+43264 = 54289 = 233^2.

(A129640(1), a(2)) = (0, 313) is a solution: 0^2+(0+313)^2 = 97969 = 313^2.

(A129640(3), a(4)) = (464, 905) is a solution: 464^2+(464+313)^2 = 215296+603729 = 819025 = 905^2.

PROG

(PARI) {forstep(n=-108, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969, &k), print1(k, ", ")))}

CROSSREFS

Cf. A129640, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160575 (decimal expansion of (363+130*sqrt(2))/313), A160576 (decimal expansion of (119187+47998*sqrt(2))/313^2).

Sequence in context: A140033 A142182 A105981 * A087862 A141280 A097446

Adjacent sequences:  A160571 A160572 A160573 * A160575 A160576 A160577

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Jun 08 2009

STATUS

approved

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Last modified December 5 23:34 EST 2019. Contains 329782 sequences. (Running on oeis4.)