A159844 -id:A159844 - cp's OEIS frontend

A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

0, 40, 901, 1077, 1281, 6160, 7180, 8364, 36777, 42721, 49621, 215220, 249864, 290080, 1255261, 1457181, 1691577, 7317064, 8493940, 9860100, 42647841, 49507177, 57469741, 248570700, 288549840, 334959064, 1448777077, 1681792581

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+359, y).
Corresponding values y of solutions (x, y) are in A159844.
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) = (363+38*sqrt(2))/359 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 0.

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

Cf. A159844, A028871, A118337, A130609, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+718*n+128881), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+718 for n > 6; a(1)=0, a(2)=40, a(3)=901, a(4)=1077, a(5)=1281, a(6)=6160.
G.f.: x*(40+861*x+176*x^2-36*x^3-287*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 359*A001652(k) for k >= 0.

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

A159845 Decimal expansion of (363 + 38*sqrt(2))/359.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A130610.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A159844.

			(363 + 38*sqrt(2))/359 = 1.16083597596149752605...

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

Cf. A130610, A159844, A002193 (decimal expansion of sqrt(2)), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

(363 +38*Sqrt(2))/359; // G. C. Greubel, May 19 2018

RealDigits[(363 +38*sqrt(2))/359, 10, 100][[1]] (* G. C. Greubel, May 19 2018 *)

(363 +38*sqrt(2))/359 \\ G. C. Greubel, May 19 2018

Equals (19 + sqrt(2))/(19 - sqrt(2)).

A159846 Decimal expansion of (293619 + 186550*sqrt(2))/359^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A130610.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159844.

			(293619 + 186550*sqrt(2))/359^2 = 4.32523444154453240860...

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

Cf. A130610, A159844, A002193 (decimal expansion of sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359).

(293619 +186550*Sqrt(2))/359^2; // G. C. Greubel, May 19 2018

RealDigits[(293619+186550Sqrt[2])/359^2,10,120][[1]] (* Harvey P. Dale, Jun 14 2011 *)

(293619 +186550*sqrt(2))/359^2 \\ G. C Greubel, May 19 2018

Equals (650 + 287*sqrt(2))/(650 - 287*sqrt(2)).

Equals (3 + 2*sqrt(2))*(19 - sqrt(2))^2/(19 + sqrt(2))^2.

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A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

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A159845 Decimal expansion of (363 + 38*sqrt(2))/359.

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A159846 Decimal expansion of (293619 + 186550*sqrt(2))/359^2.

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