A160043 -id:A160043 - cp's OEIS frontend

A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

0, 44, 95, 219, 455, 744, 1460, 2832, 4515, 8687, 16683, 26492, 50808, 97412, 154583, 296307, 567935, 901152, 1727180, 3310344, 5252475, 10066919, 19294275, 30613844, 58674480, 112455452, 178430735, 341980107, 655438583, 1039970712, 1993206308, 3820176192

Offset: 1

Mohamed Bouhamida, May 26 2007

nonn

Also values x of Pythagorean triples (x, x+73, y).
Corresponding values y of solutions (x, y) are in A160041.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A160041, A129288, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)))); // G. C. Greubel, May 07 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+73)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,44,95,219,455,744,1460},70] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

{forstep(n=0, 100000000, [3 ,1], if(issquare(2*n^2+146*n+5329), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +146 for n > 6; a(1)=0, a(2)=44, a(3)=95, a(4)=219, a(5)=455, a(6)=744.
G.f.: x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 73*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 04 2009

A160041 Positive numbers y such that y^2 is of the form x^2+(x+73)^2 with integer x.

Original entry on oeis.org

53, 73, 125, 193, 365, 697, 1105, 2117, 4057, 6437, 12337, 23645, 37517, 71905, 137813, 218665, 419093, 803233, 1274473, 2442653, 4681585, 7428173, 14236825, 27286277, 43294565, 82978297, 159036077, 252339217, 483632957, 926930185

Offset: 1

Klaus Brockhaus, May 04 2009

nonn

(-28, a(1)) and (A129289(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 1.

			(-28, a(1)) = (-28, 53) is a solution: (-28)^2+(-28+73)^2 = 784+2025 = 2809 = 53^2.
(A129289(1), a(2)) = (0, 73) is a solution: 0^2+(0+73)^2 = 5329 = 73^2.
(A129289(3), a(4)) = (95, 193) is a solution: 95^2+(95+73)^2 = 9025+28224 = 37249 = 193^2.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129289, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

I:=[53,73,125,193,365,697]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018

LinearRecurrence[{0,0,6,0,0,-1}, {53,73,125,193,365,697}, 50] (* G. C. Greubel, Apr 21 2018 *)

{forstep(n=-28, 10000000, [3, 1], if(issquare(2*n^2+146*n+5329, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1 -6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=53, a(2)=73, a(3)=125, a(4)=193, a(5)=365, a(6)=697.
G.f.: (1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 73*A001653(k) for k >= 1.

A160042 Decimal expansion of (89+36*sqrt(2))/73.

Original entry on oeis.org

1, 9, 1, 6, 5, 9, 8, 4, 6, 9, 1, 1, 5, 4, 9, 8, 9, 2, 8, 1, 7, 6, 1, 7, 5, 2, 6, 1, 2, 5, 4, 0, 9, 7, 6, 8, 2, 5, 8, 2, 3, 0, 3, 9, 3, 8, 5, 4, 2, 0, 5, 6, 5, 8, 4, 0, 3, 2, 3, 3, 5, 2, 1, 3, 2, 5, 5, 7, 0, 3, 6, 8, 8, 0, 0, 8, 7, 1, 0, 3, 2, 0, 5, 2, 8, 9, 5, 8, 2, 3, 6, 1, 0, 6, 7, 8, 2, 1, 4, 5, 4, 5, 8, 3, 6

Offset: 1

Klaus Brockhaus, May 04 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129289.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160041.

			(89+36*sqrt(2))/73 = 1.91659846911549892817...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129289, A160041, A002193 (decimal expansion of sqrt(2)), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

(89+36*Sqrt(2))/73; // G. C. Greubel, Apr 15 2018

RealDigits[(89+36Sqrt[2])/73,10,120][[1]] (* Harvey P. Dale, Jun 19 2011 *)

(89+36*sqrt(2))/73 \\ G. C. Greubel, Apr 15 2018

Equals (9+2*sqrt(2))/(9-2*sqrt(2)).

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A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

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A160041 Positive numbers y such that y^2 is of the form x^2+(x+73)^2 with integer x.

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A160042 Decimal expansion of (89+36*sqrt(2))/73.

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