cp's OEIS Frontend

A157348 Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x.

%I A157348 #11 Apr 18 2024 11:00:06
%S A157348 229,281,365,1009,1405,1961,5825,8149,11401,33941,47489,66445,197821,
%T A157348 276785,387269,1152985,1613221,2257169,6720089,9402541,13155745,
%U A157348 39167549,54802025,76677301,228285205,319409609,446908061,1330543681
%N A157348 Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x.
%C A157348 (-60, a(1)) and (A129626(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.
%H A157348 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A157348 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=229, a(2)=281, a(3)=365, a(4)=1009, a(5)=1405, a(6)=1961.
%F A157348 G.f.: x*(1-x)*(229+510*x+875*x^2+510*x^3+229*x^4) / (1-6*x^3+x^6).
%F A157348 a(3*k-1) = 281*A001653(k) for k >= 1.
%F A157348 Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
%F A157348 Limit_{n -> oo} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {0, 2}.
%F A157348 Limit_{n -> oo} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 1.
%e A157348 (-60, a(1)) = (-60, 229) is a solution: (-60)^2+(-60+281)^2 = 3600+48841 = 52441 = 229^2.
%e A157348 (A129626(1), a(2)) = (0, 281) is a solution: 0^2+(0+281)^2 = 78961 = 281^2.
%e A157348 (A129626(3), a(4)) = (559, 1009) is a solution: 559^2+(559+281)^2 = 312481+705600 = 1018081 = 1009^2.
%o A157348 (PARI) {forstep(n=-60, 200000000, [3, 1], if(issquare(2*n^2+562*n+78961, &k), print1(k, ",")))}
%Y A157348 Cf. A129626, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).
%K A157348 nonn,easy
%O A157348 1,1
%A A157348 _Klaus Brockhaus_, Apr 12 2009