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A157213
Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x.
4
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.
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
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Feb 25 2009
STATUS
approved