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%I #14 Jan 08 2022 22:16:06
%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.
%H G. C. Greubel, <a href="/A156572/b156572.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">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 a(n) = A156567(n)^2.
%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)).
%F Limit_{n -> infinity} a(n)/a(n-3) = 17 + 12*sqrt(2).
%F Limit_{n -> infinity} a(n)/a(n-1) = ((627 + 238*sqrt(2))/23^2)^2 for n mod 3 = 1.
%F Limit_{n -> infinity} a(n)/a(n-1) = ((27 + 10*sqrt(2))/23)^2 for n mod 3 = {0, 2}.
%F a(n) = -289*[n=0] + (529/4) + (3/4)*( f(n/3, 209, 5457)*(n mod 3 = 1) + f((n-1)/3, 209, 1649)*(n mod 3 = 1) + f((n-2)/2, 529, 529)*(n mod 3 = 2) ), where f(n, p, q) = p*ChebyshevU(n, 17) - q*ChebyshevU(n-1, 17). - _G. C. Greubel_, Jan 04 2022
%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, ",")))}
%o (Sage)
%o def f(n,p,q): return p*chebyshev_U(n,17) - q*chebyshev_U(n-1,17)
%o def a(n):
%o if (n%3==0): return -289*bool(n==0) + (1/4)*(529 + 3*f(n/3, 209, 5457))
%o elif (n%3==1): return (1/4)*(529 + 3*f((n-1)/3, 209, 1649))
%o else: return (1/4)*(529 + 3*f((n-2)/3, 529, 529))
%o [a(n) for n in (1..30)] # _G. C. Greubel_, Jan 04 2022
%Y Cf. A156567, 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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