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A156159
Squares of the form k^2+(k+17)^2 with integer k.
4
169, 289, 625, 2809, 7225, 18769, 93025, 243049, 635209, 3157729, 8254129, 21576025, 107267449, 280395025, 732947329, 3643933225, 9525174409, 24898630849, 123786459889, 323575532569, 845820499225, 4205095700689
OFFSET
1,1
COMMENTS
Square roots of k^2+(k+17)^2 are in A155923, values k (except for -5) are in A118120.
FORMULA
a(n) = 34*a(n-3)-a(n-6)-2312 for n > 6; a(1)=169, a(2)=289, a(3)=625, a(4)=2809, a(5)=7225, a(6)=18769.
G.f.: x*(169+120*x+336*x^2-3562*x^3+336*x^4+120*x^5+169*x^6)/((1-x)*(1-34*x^3+x^6)).
Limit_{n -> oo} a(n)/a(n-3) = (17+12*sqrt(2)).
Limit_{n -> oo} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = ((387+182*sqrt(2))/17^2)^2 for n mod 3 = 1.
EXAMPLE
625 = 25^2 is of the form k^2+(k+17)^2 with k = 7: 7^2+24^2 = 625. Hence 625 is in the sequence.
MATHEMATICA
LinearRecurrence[{1, 0, 34, -34, 0, -1, 1}, {169, 289, 625, 2809, 7225, 18769, 93025}, 30] (* Harvey P. Dale, Apr 22 2022 *)
PROG
(PARI) {forstep(n=-5, 1600000, [1, 3], if(issquare(a=2*n*(n+17)+289), print1(a, ", ")))}
CROSSREFS
Equals A155923^2. Cf. A156160 (first trisection), A156161 (second trisection), A156162 (third trisection).
Cf. A118120, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17), A157649 (decimal expansion of (387+182*sqrt(2))/17^2).
Sequence in context: A287391 A325421 A296304 * A099011 A330276 A351337
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Feb 09 2009
EXTENSIONS
G.f. corrected, fourth comment and cross-references edited by Klaus Brockhaus, Sep 23 2009
STATUS
approved