A156159 Squares of the form k^2+(k+17)^2 with integer k.

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.

Links

Table of n, a(n) for n=1..22.

Index entries for linear recurrences with constant coefficients, signature (1,0,34,-34,0,-1,1).

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

Adjacent sequences: A156156 A156157 A156158 * A156160 A156161 A156162

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