A160206 -id:A160206 - cp's OEIS frontend

A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

0, 235, 1696, 2571, 3796, 12075, 17140, 24255, 72468, 101983, 143448, 424447, 596472, 838147, 2475928, 3478563, 4887148, 14432835, 20276620, 28486455, 84122796, 118182871, 166033296, 490305655, 688822320, 967715035, 2857712848

Offset: 1

Mohamed Bouhamida, Jun 03 2007

Also values x of Pythagorean triples (x, x+857, y).
Corresponding values y of solutions (x, y) are in A160206.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 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. A160206, A001652, A123654, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5)/((1-x)*(1-6*x^3+x^6))) );  // G. C. Greubel, May 03 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,235,1696,2571,3796,12075, 17140}, 30] (* or *) CoefficientList[Series[x (235+1461x+875x^2-185x^3- 487x^4- 185x^5)/((1-x)(1-6x^3+x^6)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1714 for n > 6; a(1)=0, a(2)=235, a(3)=1696, a(4)=2571, a(5)=3796, a(6)=12075.
G.f.: x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 857*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160207 Decimal expansion of (907+210*sqrt(2))/857.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129857.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160206.

			(907+210*sqrt(2))/857 = 1.40488313663751453937...

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

Cf. A129857, A160206, A002193 (decimal expansion of sqrt(2)), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

(907+210*Sqrt(2))/857; // G. C. Greubel, Apr 25 2018

RealDigits[(907+210Sqrt[2])/857,10,150][[1]]  (* Harvey P. Dale, Mar 20 2011 *)

(907+210*sqrt(2))/857 \\ G. C. Greubel, Apr 25 2018

Equals (42+5*sqrt(2))/(42-5*sqrt(2)).

A160208 Decimal expansion of (1208787 +678878*sqrt(2))/857^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129857.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160206.

			(1208787 +678878*sqrt(2))/857^2 = 2.95305116461009821041...

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

Cf. A129857, A160206, A002193 (decimal expansion of sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857).

(1208787 +678878*Sqrt(2))/857^2; // G. C. Greubel, Apr 08 2018

RealDigits[(1208787+678878Sqrt[2])/857^2,10,120][[1]] (* Harvey P. Dale, Nov 09 2012 *)

(1208787 +678878*sqrt(2))/857^2 \\ G. C. Greubel, Apr 08 2018

Equals (1394 +487*sqrt(2))/(1394 -487*sqrt(2));

Equals (3+2*sqrt(2))*(42-5*sqrt(2))^2/(42+5*sqrt(2))^2.

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A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

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A160207 Decimal expansion of (907+210*sqrt(2))/857.

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A160208 Decimal expansion of (1208787 +678878*sqrt(2))/857^2.

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