A159574 -id:A159574 - cp's OEIS frontend

A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

0, 27, 888, 1011, 1148, 6027, 6740, 7535, 35948, 40103, 44736, 210335, 234552, 261555, 1226736, 1367883, 1525268, 7150755, 7973420, 8890727, 41678468, 46473311, 51819768, 242920727, 270867120, 302028555, 1415846568, 1578730083

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+337, y).
Corresponding values y of solutions (x, y) are in A159574.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 0.

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

Cf. A159574, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576 (decimal expansion of (278307+179662*sqrt(2))/337^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,27,888,1011,1148,6027,6740},40] (* Harvey P. Dale, Feb 26 2015 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ",")))}

a(n)=6*a(n-3)-a(n-6)+674 for n > 6; a(1)=0, a(2)=27, a(3)=888, a(4)=1011, a(5)=1148, a(6)=6027.
G.f.: x*(27+861*x+123*x^2-25*x^3-287*x^4-25*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 337*A001652(k) for k >= 0.
a(0)=0, a(1)=27, a(2)=888, a(3)=1011, a(4)=1148, a(5)=6027, a(6)=6740, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Feb 26 2015

Edited and two terms added by Klaus Brockhaus, Apr 16 2009

A159575 Decimal expansion of (339+26*sqrt(2))/337.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129999.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A159574.

			(339+26*sqrt(2))/337 = 1.11504318285371059723...

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

Cf. A129999, A159574, A002193 (decimal expansion of sqrt(2)), A159576(decimal expansion of (278307+179662*sqrt(2))/337^2).

(339+26*Sqrt(2))/337; // G. C. Greubel, May 05 2018

RealDigits[(339+26*Sqrt[2])/337, 10, 100][[1]] (* G. C. Greubel, May 05 2018 *)

(339+26*sqrt(2))/337 \\ G. C. Greubel, May 05 2018

Equals (26+sqrt(2))/(26-sqrt(2)).

A159576 Decimal expansion of (278307+179662*sqrt(2))/337^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129999.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159574.

			(278307+179662*sqrt(2))/337^2 = 4.68778836692297196116...

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

Cf. A129999, A159574, A002193 (decimal expansion of sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337).

(278307+179662*Sqrt(2))/337^2; // G. C. Greubel, May 05 2018

RealDigits[(278307+179662*Sqrt[2])/337^2, 10, 100][[1]] (* G. C. Greubel, May 05 2018 *)

(278307+179662*sqrt(2))/337^2 \\ G. C. Greubel, May 05 2018

Equals (626+287*sqrt(2))/(626-287*sqrt(2)).

Equals (3+2*sqrt(2))*(26-sqrt(2))^2/(26+sqrt(2))^2.

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A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

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A159575 Decimal expansion of (339+26*sqrt(2))/337.

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A159576 Decimal expansion of (278307+179662*sqrt(2))/337^2.

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