A160574 - OEIS

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A160574

Positive numbers y such that y^2 is of the form x^2+(x+313)^2 with integer x.

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

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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.

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)=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.

MATHEMATICA

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {233, 313, 493, 905, 1565, 2725}, 30] (* Harvey P. Dale, Dec 21 2022 *)

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