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A160055 Positive numbers y such that y^2 is of the form x^2+(x+89)^2 with integer x. 4
65, 89, 149, 241, 445, 829, 1381, 2581, 4825, 8045, 15041, 28121, 46889, 87665, 163901, 273289, 510949, 955285, 1592845, 2978029, 5567809, 9283781, 17357225, 32451569, 54109841, 101165321, 189141605, 315375265, 589634701, 1102398061 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-33, a(1)) and (A129298(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 1.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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)=65, a(2)=89, a(3)=149, a(4)=241, a(5)=445, a(6)=829.

G.f.: (1-x)*(65+154*x+303*x^2+154*x^3+65*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 89*A001653(k) for k >= 1.

EXAMPLE

(-33, a(1)) = (-33, 65) is a solution: (-33)^2+(-33+89)^2 = 1089+3136 = 4225 = 65^2.

(A129298(1), a(2)) = (0, 89) is a solution: 0^2+(0+89)^2 = 7921 = 89^2.

(A129298(3), a(4)) = (120, 241) is a solution: 120^2+(120+89)^2 = 14400+43681 = 58081 = 241^2.

MATHEMATICA

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {65, 89, 149, 241, 445, 829}, 40] (* Harvey P. Dale, Feb 04 2015 *)

PROG

(PARI) {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921, &k), print1(k, ", ")))}

CROSSREFS

Cf. A129298, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

Sequence in context: A034071 A015788 A072053 * A250642 A020140 A020194

Adjacent sequences:  A160052 A160053 A160054 * A160056 A160057 A160058

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, May 04 2009

STATUS

approved

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Last modified December 11 06:55 EST 2016. Contains 279043 sequences.