cp's OEIS Frontend

A159758 Positive numbers y such that y^2 is of the form x^2+(x+79)^2 with integer x.

%I A159758 #18 Sep 08 2022 08:45:44
%S A159758 65,79,101,289,395,541,1669,2291,3145,9725,13351,18329,56681,77815,
%T A159758 106829,330361,453539,622645,1925485,2643419,3629041,11222549,
%U A159758 15406975,21151601,65409809,89798431,123280565,381236305,523383611,718531789
%N A159758 Positive numbers y such that y^2 is of the form x^2+(x+79)^2 with integer x.
%C A159758 (-16, a(1)) and (A118676(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.
%C A159758 Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A159758 Lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {0, 2}.
%C A159758 Lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 for n mod 3 = 1.
%C A159758 For the generic case x^2 + (x+p)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m >= 5, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 42. Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1)=p, b(2) = m^2 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4)= 5*p, b(5) = 5*m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - _Mohamed Bouhamida_, Sep 09 2009
%H A159758 Harvey P. Dale, <a href="/A159758/b159758.txt">Table of n, a(n) for n = 1..1000</a>
%H A159758 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A159758 a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=65, a(2)=79, a(3)=101, a(4)=289, a(5)=395, a(6)=541.
%F A159758 G.f.: (1-x)*(65+144*x+245*x^2+144*x^3+65*x^4) / (1-6*x^3+x^6).
%F A159758 a(3*k-1) = 79*A001653(k) for k >= 1.
%e A159758 (-16, a(1)) = (-16, 65) is a solution: (-16)^2 + (-16+79)^2 = 256+3969 = 4225 = 65^2.
%e A159758 (A118676(1), a(2)) = (0, 79) is a solution: 0^2 + (0+79)^2 = 6241 = 79^2.
%e A159758 (A118676(3), a(4)) = (161, 289) is a solution: 161^2 + (161+79)^2 = 25921 + 57600 = 83521 = 289^2.
%t A159758 RecurrenceTable[{a[1]==65,a[2]==79,a[3]==101,a[4]==289,a[5]==395, a[6]== 541, a[n]==6a[n-3]-a[n-6]},a[n],{n,30}] (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{65,79,101,289,395,541},30] (* _Harvey P. Dale_, Oct 03 2011 *)
%o A159758 (PARI) {forstep(n=-16, 10000000, [1, 3], if(issquare(2*n^2+158*n+6241, &k), print1(k, ",")))}
%o A159758 (Magma) I:=[65,79,101,289,395,541]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 22 2018
%Y A159758 Cf. A118676, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).
%K A159758 nonn,easy
%O A159758 1,1
%A A159758 _Klaus Brockhaus_, Apr 30 2009