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A159565 Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x. 3
221, 241, 265, 1061, 1205, 1369, 6145, 6989, 7949, 35809, 40729, 46325, 208709, 237385, 270001, 1216445, 1383581, 1573681, 7089961, 8064101, 9172085, 41323321, 47001025, 53458829, 240849965, 273942049, 311580889, 1403776469, 1596651269 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-21,a(1)) and (A129991(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 1.

LINKS

Table of n, a(n) for n=1..29.

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)=221, a(2)=241, a(3)=265, a(4)=1061, a(5)=1205, a(6)=1369.

G.f.: (1-x)*(221+462*x+727*x^2+462*x^3+221*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 241*A001653(k) for k >= 1.

EXAMPLE

(-21, a(1)) = (-21, 221) is a solution: (-21)^2+(-21+241)^2 = 441+48400 = 48841 = 221^2.

(A129993(1), a(2)) = (0, 241) is a solution: 0^2+(0+241)^2 = 58081= 241^2.

(A129993(3), a(4)) = (620, 1061) is a solution: 620^2+(620+241)^2 = 384400+741321 = 1125721 = 1061^2.

MATHEMATICA

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {221, 241, 265, 1061, 1205, 1369}, 30] (* Harvey P. Dale, Nov 21 2011 *)

PROG

(PARI) {forstep(n=-24, 50000000, [3, 1], if(issquare(2*n^2+482*n+58081, &k), print1(k, ", ")))}

CROSSREFS

Cf. A129991, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

Sequence in context: A167825 A339680 A266237 * A330281 A345512 A048931

Adjacent sequences:  A159562 A159563 A159564 * A159566 A159567 A159568

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 16 2009

STATUS

approved

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Last modified September 26 20:34 EDT 2021. Contains 347672 sequences. (Running on oeis4.)