A160577 -id:A160577 - cp's OEIS frontend

A129641 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

0, 200, 611, 1227, 2291, 4620, 8180, 14364, 27927, 48671, 84711, 163760, 284664, 494720, 955451, 1660131, 2884427, 5569764, 9676940, 16812660, 32463951, 56402327, 97992351, 189214760, 328737840, 571142264, 1102825427, 1916025531, 3328862051, 6427738620

Offset: 1

Mohamed Bouhamida, May 31 2007

Also values x of Pythagorean triples (x, x+409, y).
Corresponding values y of solutions (x, y) are in A160577.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 0.

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

Cf. A160577, A001652, A129640, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 200, 611, 1227, 2291, 4620, 8180}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+818 for n > 6; a(1)=0, a(2)=200, a(3)=611, a(4)=1227, a(5)=2291, a(6)=4620.
G.f.: x*(200+411*x+616*x^2-136*x^3-137*x^4-136*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 409*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Jun 08 2009

A160578 Decimal expansion of (473+168*sqrt(2))/409.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jun 08 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129641.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160577.

			(473+168*sqrt(2))/409 = 1.73737867598699258728...

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

Cf. A129641, A160577, A002193 (decimal expansion of sqrt(2)), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

(473 +168*Sqrt(2))/409; // G. C. Greubel, Apr 08 2018

RealDigits[(473+168Sqrt[2])/409,10,120][[1]] (* Harvey P. Dale, Mar 05 2015 *)

(473 +168*sqrt(2))/409 \\ G. C. Greubel, Apr 08 2018

Equals (42+8*sqrt(2))/(42-8*sqrt(2)).

A160579 Decimal expansion of (204819+83570*sqrt(2))/409^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jun 08 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129641.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160577.

			(204819+83570*sqrt(2))/409^2 = 1.93091162419832230335...

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

Cf. A129641, A160577, A002193 (decimal expansion of sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409).

(204819 +83570*Sqrt(2))/409^2; // G. C. Greubel, Apr 08 2018

RealDigits[(204819+83570*Sqrt[2])/409^2,10,120][[1]] (* Harvey P. Dale, Jul 16 2013 *)

(204819 +83570*sqrt(2))/409^2 \\ G. C. Greubel, Apr 08 2018

Equals (610+137*sqrt(2))/(610-137*sqrt(2)).

Equals (3+2*sqrt(2))*(42-8*sqrt(2))^2/(42+8*sqrt(2))^2.

cp's OEIS Frontend

A129641 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

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A160578 Decimal expansion of (473+168*sqrt(2))/409.

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A160579 Decimal expansion of (204819+83570*sqrt(2))/409^2.

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