A159626 - OEIS

A159626

Positive numbers y such that y^2 is of the form x^2+(x+577)^2 with integer x.

545, 577, 613, 2657, 2885, 3133, 15397, 16733, 18185, 89725, 97513, 105977, 522953, 568345, 617677, 3047993, 3312557, 3600085, 17765005, 19306997, 20982833, 103542037, 112529425, 122296913, 603487217, 655869553, 712798645, 3517381265

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OFFSET

1,1

COMMENTS

(-33,a(1)) and (A130005(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+577)^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)=545, a(2)=577, a(3)=613, a(4)=2657, a(5)=2885, a(6)=3133.

G.f.: (1-x)*(545+1122*x+1735*x^2+1122*x^3+545*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 577*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) = (579+34*sqrt(2))/577 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 1.

EXAMPLE

(-33, a(1)) = (-33, 545) is a solution: (-33)^2+(-33+577)^2 = 1089+295936 = 297025 = 545^2.

(A130005(1), a(2)) = (0, 577) is a solution: 0^2+(0+577)^2 = 332929 = 577^2.

(A130005(3), a(4)) = (1568, 2657) is a solution: 1568^2+(1568+577)^2 = 2458624+4601025 = 7059649 = 2657^2.

PROG

(PARI) {forstep(n=-36, 50000000, [3, 1], if(issquare(2*n^2+1154*n+332929, &k), print1(k, ", ")))}

CROSSREFS

Cf. A130005, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).

Sequence in context: A022048 A107512 A264948 * A020259 A184379 A374239

Adjacent sequences: A159623 A159624 A159625 * A159627 A159628 A159629

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 21 2009

STATUS

approved