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A159626 Positive numbers y such that y^2 is of the form x^2+(x+577)^2 with integer x. 3
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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

lim_{n -> infinity} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 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)=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.

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 A053327

Adjacent sequences:  A159623 A159624 A159625 * A159627 A159628 A159629

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 21 2009

STATUS

approved

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Last modified November 27 04:21 EST 2020. Contains 338677 sequences. (Running on oeis4.)