A157213 Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x.

97, 137, 277, 305, 685, 1565, 1733, 3973, 9113, 10093, 23153, 53113, 58825, 134945, 309565, 342857, 786517, 1804277, 1998317, 4584157, 10516097, 11647045, 26718425, 61292305, 67883953, 155726393, 357237733, 395656673, 907639933

Offset

1,1

Comments

(-65, a(1)) and (A129544(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

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)=97, a(2)=137, a(3)=277, a(4)=305, a(5)=685, a(6)=1565.

G.f.: x*(1-x)*(97+234*x+511*x^2+234*x^3+97*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 137*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) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 1.

Limit_{n -> oo} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}.

Example

(-65, a(1)) = (-65, 97) is a solution: (-65)^2+(-65+137)^2 = 4225+5184 = 9409 = 97^2.
(A129544(1), a(2)) = (0, 137) is a solution: 0^2+(0+137)^2 = 18769 = 137^2.
(A129544(3), a(4)) = (136, 305) is a solution: 136^2+(136+137)^2 = 18496+74529 = 93025 = 305^2.

Prog

(PARI) {forstep(n=-68, 1000000000, [3, 1], if(issquare(n^2+(n+137)^2, &k), print1(k, ", ")))}

Crossrefs

Cf. A129544, A001653, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))).

Sequence in context: A216311 A258877 A073076 * A000923 A142528 A139500

Adjacent sequences: A157210 A157211 A157212 * A157214 A157215 A157216

Keyword

nonn,easy

Author

Klaus Brockhaus, Feb 25 2009

Status

approved