A159892 -id:A159892 - cp's OEIS frontend

A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

0, 44, 1121, 1317, 1541, 7644, 8780, 10080, 45621, 52241, 59817, 266960, 305544, 349700, 1557017, 1781901, 2039261, 9076020, 10386740, 11886744, 52899981, 60539417, 69282081, 308324744, 352850640, 403806620, 1797049361, 2056565301, 2353558517, 10473972300

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+439, y).
Corresponding values y of solutions (x, y) are in A159890.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159890, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 44, 1121, 1317, 1541, 7644, 8780}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+878*n+192721), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+878 for n > 6; a(1)=0, a(2)=44, a(3)=1121, a(4)=1317, a(5)=1541, a(6)=7644.
G.f.: x*(44+1077*x+196*x^2-40*x^3-359*x^4-40*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 439*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A159890 Positive numbers y such that y^2 is of the form x^2+(x+439)^2 with integer x.

Original entry on oeis.org

401, 439, 485, 1921, 2195, 2509, 11125, 12731, 14569, 64829, 74191, 84905, 377849, 432415, 494861, 2202265, 2520299, 2884261, 12835741, 14689379, 16810705, 74812181, 85615975, 97979969, 436037345, 499006471, 571069109, 2541411889

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-40, a(1)) and (A130645(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 1.
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

			(-40, a(1)) = (-40, 401) is a solution: (-40)^2+(-40+439)^2 = 1600+159201 = 160801 = 401^2.
(A130645(1), a(2)) = (0, 439) is a solution: 0^2+(0+439)^2 = 192721 = 439^2.
(A130645(3), a(4)) = (1121, 1921) is a solution: 1121^2+(1121+439)^2 = 1256641+2433600 = 3690241 = 1921^2.

G. C. Greubel, Table of n, a(n) for n = 1..3895
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A130645, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

I:=[401,439,485,1921,2195,2509]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018

LinearRecurrence[{0,0,6,0,0,-1}, {401,439,485,1921,2195,2509}, 50] (* G. C. Greubel, May 17 2018 *)

{forstep(n=-40, 10000000, [1, 3], if(issquare(2*n^2+878*n+192721, &k), print1(k, ",")))}

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=401, a(2)=439, a(3)=485, a(4)=1921, a(5)=2195, a(6)=2509.
G.f.: (1-x)*(401+840*x+1325*x^2+840*x^3+401*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 439*A001653(k) for k >= 1.

A159891 Decimal expansion of (443+42*sqrt(2))/439.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

lim_{n -> infinity} b(n)/b(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {1, 2}, b = A130645.

lim_{n -> infinity} b(n)/b(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {0, 2}, b = A159890.

			(443+42*sqrt(2))/439 = 1.14441223147988608667...

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

Cf. A130645, A159890, A002193 (decimal expansion of sqrt(2)), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

(443 +42*Sqrt(2))/439; // G. C. Greubel, May 17 2018

RealDigits[N[(443+42*Sqrt[2])/439,300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

(443 +42*sqrt(2))/439 \\ G. C. Greubel, May 17 2018

Equals (21 +sqrt(2))/(21 -sqrt(2)).

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A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

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A159890 Positive numbers y such that y^2 is of the form x^2+(x+439)^2 with integer x.

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A159891 Decimal expansion of (443+42*sqrt(2))/439.

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