A157213 -id:A157213 - cp's OEIS frontend

A157214 Decimal expansion of 18 + 5*sqrt(2).

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}, b = A129544.

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}, b = A157213.

			18+5*sqrt(2) = 25.07106781186547524400...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Index entries for algebraic numbers, degree 2

Cf. A129544, A157213, A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18-5*sqrt(2))/(18+5*sqrt(2))).

[18 + 5*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[18+5Sqrt[2],10,150][[1]]  (* Harvey P. Dale, Mar 18 2011 *)

18 + 5*sqrt(2) \\ G. C. Greubel, Nov 28 2017

Equals 18 + 10*A010503. - R. J. Mathar, Feb 27 2009

A157215 Decimal expansion of 18 - 5*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}, b = A129544.

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}, b = A157213.

			18 - 5*sqrt(2) = 10.92893218813452475599...

G. C. Greubel, Table of n, a(n) for n = 2..5001
Index entries for algebraic numbers, degree 2

Cf. A129544, A157213, A157214 (decimal expansion of 18+5*sqrt(2)), A157216 (decimal expansion of (18-5*sqrt(2))/(18+5*sqrt(2))).

[18 - 5*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[18 - 5*Sqrt[2], 10, 50][[1]] (* G. C. Greubel, Nov 28 2017 *)

18 - 5*sqrt(2) \\ G. C. Greubel, Nov 28 2017

A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

Original entry on oeis.org

0, 115, 136, 411, 1036, 1155, 2740, 6375, 7068, 16303, 37488, 41527, 95352, 218827, 242368, 556083, 1275748, 1412955, 3241420, 7435935, 8235636, 18892711, 43340136, 48001135, 110115120, 252605155, 279771448, 641798283, 1472291068, 1630627827, 3740674852

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+137, y).
Corresponding values y of solutions (x, y) are in A157213.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 0.

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

Cf. A157213, A001652, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))), A129288, A129289, A129298.

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,115,136,411,1036,1155,2740},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+274 for n > 6; a(1)=0, a(2)=115, a(3)=136, a(4)=411, a(5)=1036, a(6)=1155.
G.f.: x*(115+21*x+275*x^2-65*x^3-7*x^4-65*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 137*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 25 2009

A157216 Decimal expansion of (18 + 5sqrt(2))/(18 - 5sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}, b = A129544.

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}, b = A157213.

			(18 + 5*sqrt(2))/(18 - 5*sqrt(2)) = 2.29400890958816463059...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. A129544, A157213, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)).

(18+5*Sqrt(2))/(18-5*Sqrt(2)) // G. C. Greubel, Jan 27 2018

With[{c=5Sqrt[2]},RealDigits[(18+c)/(18-c),10,120][[1]]] (* Harvey P. Dale, Dec 13 2011 *)

(18+5*sqrt(2))/(18-5*sqrt(2)) \\ G. C. Greubel, Jan 27 2018

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A157214 Decimal expansion of 18 + 5*sqrt(2).

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A157215 Decimal expansion of 18 - 5*sqrt(2).

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A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

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A157216 Decimal expansion of (18 + 5*sqrt(2))/(18 - 5*sqrt(2)).

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A157216 Decimal expansion of (18 + 5sqrt(2))/(18 - 5sqrt(2)).