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A157348 Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x. 3
229, 281, 365, 1009, 1405, 1961, 5825, 8149, 11401, 33941, 47489, 66445, 197821, 276785, 387269, 1152985, 1613221, 2257169, 6720089, 9402541, 13155745, 39167549, 54802025, 76677301, 228285205, 319409609, 446908061, 1330543681 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-60, a(1)) and (A129626(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 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)=229, a(2)=281, a(3)=365, a(4)=1009, a(5)=1405, a(6)=1961.

G.f.: (1-x)*(229+510*x+875*x^2+510*x^3+229*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 281*A001653(k) for k >= 1.

EXAMPLE

(-60, a(1)) = (-60, 229) is a solution: (-60)^2+(-60+281)^2 = 3600+48841 = 52441 = 229^2.

(A129626(1), a(2)) = (0, 281) is a solution: 0^2+(0+281)^2 = 78961 = 281^2.

(A129626(3), a(4)) = (559, 1009) is a solution: 559^2+(559+281)^2 = 312481+705600 = 1018081 = 1009^2.

PROG

(PARI) {forstep(n=-60, 200000000, [3, 1], if(issquare(2*n^2+562*n+78961, &k), print1(k, ", ")))}

CROSSREFS

Cf. A129626, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

Sequence in context: A094612 A250237 A112847 * A142221 A142779 A139512

Adjacent sequences:  A157345 A157346 A157347 * A157349 A157350 A157351

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 12 2009

STATUS

approved

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Last modified November 22 13:47 EST 2019. Contains 329393 sequences. (Running on oeis4.)