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A156572 Squares of the form k^2+(k+23)^2 with integer k. 3

%I

%S 289,529,1369,4225,13225,42025,139129,444889,1423249,4721929,15108769,

%T 48344209,160402225,513249025,1642275625,5448949489,17435353849,

%U 55789022809,185103876169,592288777609,1895184495649,6288082836025

%N Squares of the form k^2+(k+23)^2 with integer k.

%C Square roots of k^2+(k+17)^2 are in A156567, values k are in A118337.

%C lim_{n -> infinity} a(n)/a(n-3) = 17+12*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = ((627+238*sqrt(2))/23^2)^2 for n mod 3 = 1.

%C lim_{n -> infinity} a(n)/a(n-1) = ((27+10*sqrt(2))/23)^2 for n mod 3 = {0, 2}.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,34,-34,0,-1,1).

%F a(n) = 34*a(n-3)-a(n-6)-4232 for n > 6; a(1)=289, a(2)=529, a(3)=1369, a(4)=4225, a(5)=13225, a(6)=42025.

%F G.f.: x*(289+240*x+840*x^2-6970*x^3+840*x^4+240*x^5+289*x^6)/((1-x)*(1-34*x^3+x^6)).

%e 4225 = 65^2 is of the form k^2+(k+23)^2 with k = 33: 33^2+56^2 = 4225. Hence 4225 is in the sequence.

%t LinearRecurrence[{1,0,34,-34,0,-1,1},{289,529,1369,4225,13225,42025,139129},30] (* _Harvey P. Dale_, Mar 21 2020 *)

%o (PARI) {forstep(n=-8, 1800000, [1, 3], if(issquare(a=2*n*(n+23)+529), print1(a, ",")))}

%Y Equals A156567^2. Cf. A156575 (first trisection), A156573 (second trisection), A156574 (third trisection).

%Y Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Feb 11 2009

%E Revised by _Klaus Brockhaus_, Feb 16 2009

%E G.f. corrected, third comment and cross-references edited by _Klaus Brockhaus_, Sep 22 2009

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Last modified August 1 17:21 EDT 2021. Contains 346402 sequences. (Running on oeis4.)