A157348 -id:A157348 - cp's OEIS frontend

A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

0, 76, 559, 843, 1239, 3976, 5620, 7920, 23859, 33439, 46843, 139740, 195576, 273700, 815143, 1140579, 1595919, 4751680, 6648460, 9302376, 27695499, 38750743, 54218899, 161421876, 225856560, 316011580, 940836319, 1316389179, 1841851143, 5483596600

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+281, y).
Corresponding values y of solutions (x, y) are in A157348.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0.

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

Cf. A157348, A001652, A129288, A129289, A129298, 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).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 76, 559, 843, 1239, 3976, 5620}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 1000000000, [3, 1], if(issquare(2*n^2+562*n+78961), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+562 for n > 6; a(1)=0, a(2)=76, a(3)=559, a(4)=843, a(5)=1239, a(6)=3976.
G.f.: x*(76+483*x+284*x^2-60*x^3-161*x^4-60*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 281*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Apr 12 2009

A157349 Decimal expansion of (297 + 68*sqrt(2))/281.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2009

lim_{n -> infinity} b(n)/b(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {1, 2}, b = A129626.

lim_{n -> infinity} b(n)/b(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {0, 2}, b = A157348.

			(297 + 68*sqrt(2))/281 = 1.39916911829669204027...

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

Cf. A129626, A157348, A002193 (decimal expansion of sqrt(2)), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

(297+68*Sqrt(2))/281; // G. C. Greubel, Feb 01 2018

RealDigits[(297 + 68*Sqrt[2])/281, 10, 100][[1]] (* G. C. Greubel, Feb 01 2018 *)

(297+68*sqrt(2))/281 \\ G. C. Greubel, Feb 01 2018

(297 + 68*sqrt(2))/281 = (17 + 2*sqrt(2))/(17 - 2*sqrt(2)).

A157350 Decimal expansion of (130803 + 73738*sqrt(2))/281^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2009

lim_{n -> infinity} b(n)/b(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0, b = A129626.

lim_{n -> infinity} b(n)/b(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 1, b = A157348.

			(130803 + 73738*sqrt(2))/281^2 = 2.97722014237746840476...

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

Cf. A129626, A157348, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281).

(130803+73738*Sqrt(2))/281^2 // G. C. Greubel, Feb 01 2018

RealDigits[(130803 + 73738*Sqrt[2])/281^2, 10, 100][[1]] (* G. C. Greubel, Feb 01 2018 *)

(130803+73738*sqrt(2))/281^2 \\ G. C. Greubel, Feb 01 2018

(130803 + 73738*sqrt(2))/281^2 = (458 + 161*sqrt(2))/(458 - 161*sqrt(2)) = (3 + 2*sqrt(2))*(17 - 2*sqrt(2))^2/(17 + 2*sqrt(2))^2.

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A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

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A157349 Decimal expansion of (297 + 68*sqrt(2))/281.

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A157350 Decimal expansion of (130803 + 73738*sqrt(2))/281^2.

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